VECTOR CALCULUS FOR TAMED DIRICHLET SPACES

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VECTOR CALCULUS FOR TAMED DIRICHLET SPACES

MATHIAS BRAUN

Abstract. In the language of L∞-modules proposed by Gigli, we introducea first order calculus on a topological Lusin measure space (M ,m) carryinga quasi-regular, strongly local Dirichlet form E. Furthermore, we develop asecond order calculus if (M ,E,m) is tamed by a signed measure in the extendedKato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows usto define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, andsecond fundamental form.

Contents

0. Introduction 2List of main vector spaces 9

Part 1. First order differential structure 101. Preliminaries. Dirichlet spaces and module theory 10

1.1. Notations 101.2. Riemannian manifolds with boundary 121.3. Dirichlet forms 131.4. L∞-modules 20

2. Cotangent module 262.1. The construction 262.2. Differential of a function in the extended domain 282.3. Some remarks on the axiomatization 31

3. Tangent module 323.1. Tangent vector fields and gradients 323.2. Divergences 33

Part 2. Second order differential structure 374. Preliminaries. The taming condition 37

4.1. Basic notions of tamed spaces 374.2. Self-improvement and singular Γ2-calculus 404.3. Lebesgue spaces and test objects 44

5. Hessian 475.1. The Sobolev space D(Hess) 475.2. Existence of many functions in D(Hess) 515.3. Structural consequences of Lemma 5.9 585.4. Calculus rules 60

6. Covariant derivative 666.1. The Sobolev space W 1,2(TM ) 666.2. Calculus rules 686.3. Heat flow on vector fields 73

Date: August 30, 2021.2010 Mathematics Subject Classification. Primary: 53C21, 58J35; Secondary: 31C25, 35J10,

35K20, 58J32.Key words and phrases. Heat flow; Kato class; Ricci curvature; Schrödinger semigroup.

http://arxiv.org/abs/2108.12374v1

2 MATHIAS BRAUN

6.4. Bits of tensor calculus 787. Exterior derivative 83

7.1. The Sobolev space D(dk) 837.2. Nonsmooth de Rham cohomology and Hodge theorem 89

8. Curvature measures 938.1. κ-Ricci measure 938.2. Curvature tensors from the κ-Ricci measure 998.3. Vector Bochner inequality 1078.4. Heat flow on 1-forms 107

Appendix A. Extrinsic approaches 111A.1. Sets of finite perimeter 111A.2. Regular semiconvex subsets 113

References 114

0. Introduction

Background. The so-called RCD(K,∞) condition, K ∈ R, introduced in [3, 5, 77,93, 94], gives a meaning to the Ricci curvature of an infinitesimally Hilbertian metricmeasure space (M , d,m) being bounded from below by K. It can be defined in atleast two equivalent ways [5, 6]: the Lagrangian one by geodesic K-convexity of therelative entropy on the 2-Wasserstein space, as well as the Eulerian one phrasedas a weak version of the Bakry–Émery condition BE2(K,∞) [8, 9] coupled withthe Sobolev-to-Lipschitz property. In the latter picture, Gigli [44, 45] developed apowerful first and second order calculus. It leads to natural nonsmooth analogues tothe notions of Hessian, covariant and exterior derivative, Ricci curvature, Hodge’stheorem, etc. This machinery has already provided deep structural and geometricresults [20, 21]. In the abstract diffusion operator setting, a pointwise definition ofa Ricci tensor is due to [96].

Recently, spaces with nonconstant, even nonuniform lower Ricci bounds have at-tracted high attention. Using Schrödinger operator theory, the condition BE2(K,∞)can be given a meaning — which also works perfectly in the framework of Dirichletspaces (M ,E,m) — even if K is replaced by a function, a measure, or a distri-bution [41, 98]. (For functions, an equivalent Lagrangian counterpart still exists[17, 95, 98].) In fact, there is important evidence in the measure-valued case onceboundaries come into play. First, albeit, say, compact Riemannian manifolds withconvex boundary are still covered by the RCD theory [55], already the appearanceof a small boundary concavity makes it impossible for the relative entropy to beK-convex for any K ∈ R [98, 102]. Second, on a compact Riemannian manifold Mwith not necessarily convex boundary ∂M , the signed Borel measure

κ := k v + ls (0.1)

plays the natural role of a lower “Ricci” bound [41, 67, 98]. Here v and s are thevolume measure and surface measure of M and ∂M , and k and l are the pointwiselowest eigenvalues of the Ricci tensor Ric and the second fundamental form I.

Of particular interest in the outlined business of singular Ricci bounds is theextended Kato class K1−(M ) of signed measures on M , already for Riemannianmanifolds without boundary [15, 24, 49, 50, 51, 79, 85, 86] or their Ricci limits [25].This is just the right class of measure-valued potentials for which the associatedFeynman–Kac semigroup has good properties [92]. In a recent work, Erbar, Rigoni,Sturm and Tamanini [41] introduced the notion of tamed spaces, i.e. Dirichlet spaces

VECTOR CALCULUS FOR TAMED DIRICHLET SPACES 3

supporting distribution-valued synthetic lower Ricci bounds in terms of E-quasi-local distributions κ ∈ F

−1qloc. These include the extended Kato class of M .

Objective. Inspired by and following [45], our goal is to construct a functional firstand second order calculus over Dirichlet spaces that are tamed by signed extendedKato class measures. (There are various reasons for working with κ ∈ K1−(M )rather than with general κ ∈ F

−1qloc, which are summarized in an own paragraph

below. Still, already in the former case, many arguments become technically morechallenging compared to [45].) In turn, this will induce a first order calculus onvector-valued objects. A functional first order structure for Dirichlet spaces is, ofcourse, well-known to exist [10, 30, 36, 61, 62, 68]. Here, we introduce it by theapproach through L∞-modules [45] and show its compatibility with the previousworks. On the other hand, besides [45] higher order objects are only studied inone-dimensional cases [10, 62] or under restrictive structural assumptions [63, 76].In our general approach, the two most important quantities will be

• the Hessian operator on appropriate functions, along with proving thatsufficiently many of these do exist, and

• a measure-valued Ricci curvature.

In addition, we concisely incorporate the tamed analogue of the finite-dimensionalBE2(K,N) condition [41], K ∈ R and N ∈ [1,∞), following the RCD∗(K,N)-treatise [53] which is not essentially different from [45]. More details are outlinedbelow. Before, we summarize our motivations in extending [45] to tamed spaces.

The first, evident, reason is the larger setting. Dirichlet spaces are more generalthan metric measure spaces: they cover e.g. certain noncomplete spaces, extendedmetric measure spaces such as configuration spaces [1, 39], etc. From many perspec-tives, they seem to be the correct framework in which elements of a vector calculusshould be studied [10, 61]. Also, the considered lower Ricci bounds, examples ofwhich are due to [18, 41, 51], may be highly irregular. In fact [65], already foruniform lower bounds, the Bakry–Émery setting is strictly larger than the RCDone if the Sobolev-to-Lipschitz property is dropped. See Subsection 4.1.5 below.

Second, we want to pursue a thorough discussion of how the appearance of a“boundary” — or more precisely, an m-negligible, non-E-polar set — in M affectsthe calculus objects that are introduced similarly as in [45]. Besides the need ofmeasure-valued Ricci bounds to describe curvature of m-singular sets as by (0.1),boundaries play an increasing role in recent research [21, 98]. This motivated us tomake sense, in all generality and apart from extrinsic structures, of measure-valuedboundary objects such as normal components of vector fields or, reminiscent of [54],a second fundamental form. In fact, our guiding example is the case of compactRiemannian manifolds with boundary which, unlike only partly in the RCD setting,is fully covered by tamed spaces. Returning to this setting from time to time alsoprovides us with a negative insight on an open question in [45], namely whether“H = W”, see e.g. Section 5.4. (This does not conflict with the smooth “H = W”results [88] as our “H-spaces” are different from the smooth ones.)

Lastly, we hope that our toolbox becomes helpful in further investigations oftamed spaces. Possible directions could include

• the study of covariant Schrödinger operators [16, 50],• rigidity results for and properties of finite-dimensional tamed spaces [21, 22],• the study of bounded variation functions under Kato conditions [20, 23, 49],• super-Ricci flows [70, 97], noting that the Kato condition, in contrast to

Lp-conditions, on the Ricci curvature along Kähler–Ricci flows is stable[101],

• a structure theory for Kato Ricci limit or tamed spaces [25, 82], or

4 MATHIAS BRAUN

• the proof of a Bismut–Elworthy–Li formula [11, 38].

First order calculus. Let (M ,E,m) be a quasi-regular, strongly local Dirichlet space,see Section 1.3 for basics on these. To simplify the presentation in this introduction,we assume that E admits a carré du champ Γ: Fe → L1(M ), i.e.

E(f) =∥∥Γ(f)

∥∥L1(M)

for every f ∈ F, where F (or Fe) is the (extended) domain of E. This will beour setting from Chapter 3 on, see Assumption 3.1. However, the space of 1-formsin Chapter 2 can even be constructed in a “universal” sense, see Theorem 2.9, byrelying on the more general concepts of energy measures for E and E-dominantmeasures [33, 60], cf. Subsection 1.3.4 and Subsection 1.3.5.

To speak about vector-valued objects, we employ the theory of Lp-normed L∞-modules, p ∈ [1,∞], w.r.t. a given measure — here m — introduced in [45], seeSection 1.4. This is a Banach space M endowed with a group action by L∞(M )and a map | · | : M → Lp(M ), the pointwise norm, such that

‖ · ‖M =∥∥| · |

∥∥Lp(M)

.

In terms of | · |, all relevant m-a.e. properties of elements of M, e.g. their m-a.e.vanishing outside some given Borel set A ⊂ M , can be rigorously made sense of.L∞(M ) is chosen as acting group given that multiplying vector-valued objects byfunctions should preserve the initial object’s m-integrability. Thus, to some extentL∞-modules allow us to speak of generalized sections without any vector bundle(which we will also not define). We believe that this interpretation is more straight-forward and better suited for analytic purposes than the fiber one by measurableHilbert fields from [10, 30, 36, 61, 62, 68] — albeit the approaches are equivalent,see Remark 1.30 — where such a bundle is actually constructed.

The space L2(T ∗M ) of L2-1-forms w.r.t. m, termed cotangent module [45], isexplicitly constructed in Section 2.1. By duality, the tangent module L2(TM ) ofL2-vector fields w.r.t. m is then defined in Section 3.1. All in all, the discussionfrom Chapter 2 and Chapter 3 leads to the following.

Theorem 0.1. L2(T ∗M ) and L2(TM ) are L2-normed L∞-modules with pointwisenorms both denoted by | · |. They come with a linear differential d: Fe → L2(T ∗M )and a linear gradient ∇ : Fe → L2(TM ) such that for every f ∈ Fe,

|df | = |∇f | = Γ(f)1/2m-a.e.

Both d and ∇ obey all expected locality and calculus rules, cf. Proposition 2.11.Moreover, polarization of | · | induces a pointwise scalar product 〈·, ·〉 on L2(T ∗M )2

and L2(TM )2 which, by integration w.r.t. m, turns the latter into Hilbert spaces,respectively, see Theorem 2.4.

Measure-valued divergence. Recall the Gauß–Green formula

−ˆ

M

dh(X) dv =ˆ

M

h divvX dv −ˆ

∂M

h 〈X, n〉 ds, (0.2)

valid for every compact Riemannian manifold M with boundary ∂M , every X ∈Γc(TM ) and every h ∈ C∞

c (M ). Here, n is the outward-pointing unit normalvector field at ∂M . This motivates our first key differential object, the measure-valued divergence of appropriate vector fields, which in turn is suitable to define thenormal component of the latter. Indeed, the point of introducing these, and theessence of our “boundary discussion”, is that the second fundamental form of ∂Mat gradients is — loosely speaking, see Section 1.2 and Example 8.24 — given by

I(∇f,∇f) = −12

⟨∇|∇f |2, n

⟩(0.3)


for every f ∈ C∞(M ) for which ∇f ∈ Γ(TM ) is tangential to ∂M , i.e.

〈∇f, n〉 = 0 at ∂M . (0.4)

We thus propose the following definition leaned on [23], see Definition 3.7.

Definition 0.2. We say that X ∈ L2(TM ) has a measure-valued divergence, brieflyX ∈ D(div), if there exists a σ-finite signed Borel measure divX charging no E-polar sets such that for sufficiently many h ∈ F,

−ˆ

M

dh(X) dm =ˆ

M

h ddivX.

In turn, keeping in mind (0.2) and using Lebesgue’s decomposition

divX = div≪ X + div⊥ X

of divX w.r.t. m, we define the normal component of X ∈ D(div) by

nX := − div⊥ X,

see Definition 3.8. Calculus rules for divX and nX , X ∈ D(div), are listed inSection 3.2. In our generality, we do not know more about the support of nXthan its m-singularity. Nevertheless, these notions are satisfactorily compatiblewith other recent extrinsic approaches to Gauß–Green’s formula and boundarycomponents on (subsets of) RCD spaces [20, 23, 98] as outlined in Appendix A.

The advantage of this measure point of view compared to the L2-one from [45],see Definition 3.5, is its ability to “see” the normal component of X ∈ D(div)rather than the latter being left out in the relevant integration by parts formulasand interpreted as zero. This distinction does mostly not matter: matching withthe interpretation of the generator ∆ of E as Neumann Laplacian, on tamed spaces,for many g ∈ F∩ L∞(M ) and f ∈ D(∆) — e.g. for g, f ∈ Test(M ), cf. Lemma 3.15and (0.7) below — the vector field X := g∇f ∈ L2(TM ) belongs to D(div) with

div≪ X =[dg(∇f) + g∆f

]m,

nX = 0.(0.5)

In particular, reminiscent of (0.3) and (0.4) one would desire to have a large classof vector fields with vanishing normal component to define a second fundamentalform. (In fact, many relevant spaces will be defined in terms of such vector fields,hence all Laplace-type operators considered in this work, see Definition 6.20 andDefinition 7.21, implicitly obey Neumann boundary conditions in certain senses.)By now, it is however not even clear if there exist (m)any f ∈ Fwith

a. |∇f |2 ∈ F, not to say withb. ∇|∇f |2 ∈ D(div).

These issues appear similarly when initially trying to define higher order differentialoperators, as briefly illustrated now along with addressing a. and b.

Second order calculus. The subsequent pointwise formulas hold on the interior M

of any Riemannian manifold M with boundary, for every f, g1, g2 ∈ C∞(M ), everyX,X1, X2 ∈ Γ(TM ) and every ω ∈ Γ(T ∗M ) [74, 84]:

2 Hess f(∇g1,∇g2) =⟨∇〈∇f,∇g1〉,∇g2

⟩+

⟨∇〈∇f,∇g2〉,∇g1

⟩

−⟨∇〈∇g1,∇g2〉,∇f

⟩,

⟨∇∇g1X,∇g2

⟩=

⟨∇〈X,∇g1〉,∇g2

⟩− Hess g2(X,∇g1),

dω(X1, X2) = d[ω(X2)

](X1) − d

[ω(X1)

](X2)

− ω(∇X1X2 − ∇X2X1).∣∣

(0.6)

6 MATHIAS BRAUN

The first identity characterizes the Hessian Hess f of f , the second is a definitionof the covariant derivative ∇X of X in terms of that Hessian, and in turn, theexterior derivative dω of ω can be defined with the help of ∇. (A similar formulais true for the exterior differential acting on forms of any degree, see Example 7.1.)Hence, we may and will axiomatize these three differential operators in the previousorder. In the sequel, we only outline how we paraphrase the first identity in (0.6)nonsmoothly. The operators ∇ and d can then be defined by similar (integrationby parts) procedures and, as for the Hessian, satisfy a great diversity of expectedcalculus rules, see Section 5.4, Chapter 6 and Chapter 7 for details.

Up to the small point of defining the two-fold tensor product L2((T ∗)⊗2M ) ofL2(T ∗M ), see Subsection 1.4.5, and keeping in mind (0.5), the following, stated inDefinition 5.2, is naturally motivated by (0.6).

Definition 0.3. The space D(Hess) consists of all f ∈ F such that there existssome tensor Hess f ∈ L2((T ∗)⊗2M ) such that for every g1, g2 ∈ Test(M ),

2ˆ

M

hHess f(∇g1,∇g2) dm

= −ˆ

M

〈∇f,∇g1〉 ddiv≪(h∇g2) −ˆ

M

〈∇f,∇g2〉 ddiv≪(h∇g1)

−ˆ

M

h⟨∇f,∇〈∇g1,∇g2〉

⟩dm.

The advantage of Definition 0.3 is that the r.h.s. of the defining property onlycontains one derivative of f . All terms make sense if, as stated, g1 and g2 are in

Test(M ) :=f ∈ D(∆) ∩ L∞(M ) : |∇f | ∈ L∞(M ), ∆f ∈ F

, (0.7)

cf. Subsection 4.2.1 and Section 5.1. Test(M ) is dense in F, and is a cornerstoneof our discussion, playing the role of smooth functions. For instance, |∇f |2 ∈ F forf ∈ Test(M ) by Proposition 4.20, which also addresses a. above. (In fact, |∇f |2is in the domain of the measure-valued Schrödinger operator ∆2κ, Definition 4.19.For possible later extensions, κ will mostly not be separated from the consideredoperators. Hence, b. will be answered quite late, but positively, in Lemma 8.12.)The latter technical grounds have been laid in [41] following [87], are summarizedin Section 4.2, and are one key place where taming by κ ∈ K1−(M ) is needed.

In Theorem 5.11, we show that D(Hess) is nonempty, in fact, dense in L2(M ).

Theorem 0.4. Every f ∈ Test(M ) belongs to D(Hess) withˆ

M

∣∣Hess f∣∣2

HSdm ≤

ˆ

M

(∆f)2 dm −⟨κ

∣∣ |∇f |2⟩.

Here | · |HS is the pointwise Hilbert–Schmidt-type norm on L2((T ∗)⊗2M ) — aswell as the two-fold tensor product L2(T⊗2M ) of L2(TM ) — see Subsection 1.4.5.

The key ingredient for the proof of Theorem 0.4 is Lemma 5.9. It results from avariant of the famous self-improvement technique [8]. Here we follow [45], see also[41, 87, 96]. The idea is to replace f ∈ Test(M ) in the taming condition

∆2κ |∇f |22

−⟨∇f,∇∆f

⟩m ≥ 0

from Proposition 4.20 by a polynomial in appropriate test functions. By optimizingover the coefficients, Theorem 0.4 follows by integrating the resulting inequality

∆2κ |∇f |22

−⟨∇f,∇∆f

⟩m ≥

∣∣Hess f∣∣2

HSm.


Ricci curvature and second fundamental form. The second main result of our workis the existence of the named measure-valued curvature tensors. Both are definedby Bochner’s identity. The latter requires some work to be made sense of at leastfor the large class Reg(TM ) of regular vector fields, i.e. all linear combinations ofelements of the form X := g∇f ∈ L2(TM ), g ∈ Test(M ) ∪ R 1M and f ∈ Test(M ),see Section 4.3. (It is generally larger than the one of test vector fields Test(TM )considered in [45].) Such X , first, obey |X |2 ∈ D(∆2κ) by Lemma 8.2, second, havea covariant derivative ∇X ∈ L2(T⊗2M ) by Theorem 6.3, and third, have a 1-formcounterpart X ∈ L2(T ∗M ) in the domain of the Hodge Laplacian ~∆ by Lemma 8.1.Therefore, for X ∈ Reg(TM ) the definition

Ricκ(X,X) := ∆2κ |X |22

+ ~∆X(X)m −∣∣∇X

∣∣2

HSm

makes sense. In fact, a variant of which is Theorem 8.9, we have the following.

Theorem 0.5. The previous map Ricκ extends continuously to the closure H1,2♯ (TM )

of Reg(TM ) w.r.t. an appropriate H1,2-norm, see Definition 8.7, with values in thespace of Borel measures on M with finite total variation charging no E-polar sets.

The nonnegativity implicitly asserted therein comes precisely from the tamingcondition. Abusing terminology, the map Ricκ will be called κ-Ricci measure.

Finally, in Section 8.2 we separate the measure κ from Ricκ. To this aim, inLemma 8.2 and Lemma 8.12 we discover that ∇|X |2 ∈ D(div) together with therelation div ∇|X |2 = ∆2κ|X | + 2 |X |2∼ κ — for an E-quasi-continuous m-version|X |2∼ of |X |2 — for every X ∈ Reg(TM ), linking the operator div to the κ-Riccimeasure Ricκ (recall b. above). Based on this observation we then set

Ric(X,X) := Ricκ(X,X) + |X |2∼ κ

: = div ∇|X |22

+ ~∆X(X)m −∣∣∇X

∣∣2

HSm

(0.8)

for X ∈ Reg(TM ), which in turn allows us to define, even for X ∈ H1,2♯ (TM ),

• the Ricci curvature of M , see Definition 8.16, by Ric≪(X,X), and• the second fundamental form of M , see Definition 8.23, by

II(X,X) := Ric⊥(X,X).

These definitions have serious smooth evidence thanks to — and in fact have beenpartly inspired by — the work [54]. Therein, it is shown that on any Riemannianmanifold M with boundary, the map Ric similarly defined according to [45], henceto our work as well, satisfies

Ric(∇f,∇f) = Ric(∇f,∇f) v + I(∇f,∇f) s

for every f ∈ C∞c (M ) subject to (0.4). Moreover, on RCD(K,∞) spaces, K ∈ R,

according to [45] we particularly have II(X,X) ≥ 0 for every X ∈ H1,2♯ (TM ), which

is a way of analytically stating convexity of M , see Remark 8.26. A further novel,but natural suggestion of our treatise is to interpret the Ricci curvature of tamedspaces as something m-absolutely continuous.

Other interesting results. Our treatise comes with further beautiful results that areworth mentioning here and hold in great generality. Examples are

• metric compatibility of the covariant derivative ∇ w.r.t. the “Riemannianmetric” 〈·, ·〉, Proposition 6.11, and

• a nonsmooth analogue of the Hodge theorem, Theorem 7.23.

Moreover, we address various points that have not been treated in [45], but ratherinitiated in [16, 53], among others

8 MATHIAS BRAUN

• semigroup domination of the heat flow on vector fields w.r.t. the func-tional one, Theorem 6.26, as well as of the heat flow on 1-forms w.r.t. theSchrödinger semigroup with potential κ, Theorem 8.40,

• spectral bottom estimates for the Bochner Laplacian, Corollary 6.21, andthe Hodge Laplacian, Corollary 8.35,

• a vector version of the measure-valued q-Bochner inequality [17], q ∈ [1, 2],Theorem 8.28, and

• boundedness of the “local dimension” of L2(TM ) by ⌊N⌋, Proposition 5.14,and the existence of an extension of Ric≪ to all of L2(TM ) on tamed spaceswith upper dimension bound N ∈ [1,∞), Theorem 8.21.

Comments on the extended Kato condition. Finally, we comment on the assumptionκ ∈ K1−(M ) and technical issues, compared to [45], which arise later.

In [45, Cor. 3.3.9, Cor. 3.6.4], the following “integrated Bochner inequality” forRCD(K,∞) spaces, K ∈ R, is derived for suitable X ∈ L2(TM ):

ˆ

M

∣∣∇X∣∣2

HSdm ≤

ˆ

M

|dX|2 dm +ˆ

M

|δX|2 dm −K

ˆ

M

|X |2 dm. (0.9)

Here δ is the codifferential operator. The interpretation of (0.9) is that an appro-priate first order norm on 1-forms controls the first order topology on vector fieldsqualitatively and quantitatively. Indeed, first, for gradient vector fields, by heatflow regularization (0.9) implies that D(∆) ⊂ D(Hess), and (0.9) is stable underthis procedure. Second, (0.9) is crucial in the RCD version of Theorem 0.5 [45,Thm. 3.6.7], for extending (0.8) beyond Test(TM ) requires continuous dependencyof the covariant term w.r.t. a contravariant norm. In both cases, the curvature termis clearly continuous, even in F or L2(TM ), respectively.

The latter is wrong in our situation: already on a compact Riemannian manifoldM with boundary and l 6= 0, the pairing

⟨κ

∣∣ |X |2⟩

according to (0.1) does noteven make sense for general X ∈ L2(TM ). Hence, we will have to deal with twocorrelated problems: controlling our calculus by stronger continuity properties ofX 7→

⟨κ

∣∣ |X |2⟩, but also vice versa. (The fact that certain first order norms on

1-forms bound covariant ones on compact Riemannian manifolds with boundary, aclassical result by Gaffney [88], see Remark 7.20, is already nontrivial.)

The key property of κ ∈ K1−(M ) in this direction is that f 7→⟨κ

∣∣ f2⟩

is (well-defined and) E-form bounded on Fwith form bound smaller than 1, see Lemma 4.5.That is, there exist ρ′ ∈ [0, 1) and α′ ∈ R such that for every f ∈ F,

∣∣⟨κ∣∣ f2

⟩∣∣ ≤ ρ′

ˆ

M

|∇f |2 dm + α′

ˆ

M

f2 dm. (0.10)

Now, from (0.10), we first note that the pairing⟨κ

∣∣ |X |2⟩

is well-defined for allX in a covariant first order space termed H1,2(TM ), see Definition 6.5, since forevery X ∈ H1,2(TM ) we have |X | ∈ F by Kato’s inequality

∣∣∇|X |∣∣ ≤

∣∣∇X∣∣HS

m-a.e., (0.11)

as proven in Lemma 6.13. The latter is essentially a consequence of metric compat-ibility of ∇, cf. Proposition 6.11, and Cauchy–Schwarz’s inequality. In particular,combining (0.10) with (0.11) will imply that X 7→

⟨κ

∣∣ |X |2⟩

is even continuous inH1,2(TM ), see Corollary 6.14. For completeness, we also mention here that Kato’sinequality is useful at other places as well, e.g. in proving the above mentionedsemigroup domination results. On RCD(K,∞) spaces, K ∈ R — on which (0.11)has been proven in [32] in order to find “quasi-continuous representatives” of vectorfields — this has been observed in [16].


However, extending the inequalityˆ

M

∣∣∇X∣∣2

HSdm ≤

ˆ

M

|dX|2 dm +ˆ

M

|δX|2 dm −⟨κ

∣∣ |X |2⟩

(0.12)

similar to (0.9), see Lemma 8.8, from X ∈ Reg(TM ) — for which it is valid bymany careful computations, see Lemma 5.9 and Lemma 8.2, and the BE1(κ,∞)condition, see Proposition 4.24 and Corollary 8.6 — continuously to more generalX ∈ H1,2

♯ (TM ) requires better control on the curvature term. Here is where theform bound ρ′ ∈ [0, 1) comes into play. Indeed, using (0.10) and (0.11),

−⟨κ

∣∣ |X |2⟩

≤ ρ′

ˆ

M

∣∣∇X∣∣2

HSdm + α′

ˆ

M

|X |2 dm,

and this can be merged with (0.12) to obtain the desired continuous control of thecovariant by a contravariant first order norm. In fact, this kind of argumentation,without already having Kato’s inequality at our disposal, will also be pursued inour proof that D(∆) ⊂ D(Hess), see Corollary 5.12.

In view of this key argument, we believe that the extended Kato frameworkis somewhat maximal possible for which a second order calculus, at least with thepresented diversity of higher order differential operators, as below can be developed.

Lastly, it is worth to spend few words on a different technical issue. Namely, tocontinuously extend Ricκ in Theorem 0.5 w.r.t. a meaningful target topology, weneed to know in advance that ∆2κ|X |2 has finite total variation for X ∈ Reg(TM ).Even for gradient vector fields, this is not discussed in [41]. On the other hand, thecorresponding RCD space result [87, Lem. 2.6] uses their stochastic completeness [3].In our work, the latter is neither assumed nor generally known to be a consequenceof the condition κ ∈ K1−(M ). Compare with Subsection 4.1.6. In Proposition 4.24,we give an alternative, seemingly new proof of the above finiteness which reliesinstead on the BE1(κ,∞) condition.

Organization. Part 1 deals with the first order differential calculus over M . Wefirst recapitulate basic notions about Dirichlet forms and L∞-modules in Chapter 1.Then appropriate analogs of differential 1-forms and vector fields are introduced inChapter 2 and Chapter 3, respectively. The latter contains a thorough discussionon functional and measure-valued divergences in Section 3.2.

In Part 2, we study elements of a second order calculus on tamed spaces, whoseproperties are recorded in Section 4.1. We go on with giving a meaning to the Hes-sian in Chapter 5, to the covariant derivative in Chapter 6, and exterior differentialand Hodge theory in Chapter 7. Chapter 8 contains the appropriate existence of aRicci curvature and a second fundamental form.

Acknowledgments. The author warmly thanks Lorenzo Dello Schiavo, Chiara Rigoni,Christian Rose, Karl-Theodor Sturm and Luca Tamanini for many fruitful and en-lightening discussions as well as helpful comments.

Funding by the European Research Council through the ERC-AdG “RicciBounds”is gratefully acknowledged.

List of main vector spaces

The page numbers refer to the places in the text where the respective objectsare introduced first.

F, D(E) domain of E, p. 13Fe extended domain, p. 14

C0(M ) continuous functions vanishing at ∞, p. 19D(∆) domain of the Laplacian, p. 17

D(∆qκ) domain of the Schrödinger operator with

10 MATHIAS BRAUN

potential qκ, p. 39D(∆) domain of the measure-valued Laplacian, p. 99

D(∆qκ) domain of the measure-valued Schrödingeroperator with potential qκ, p. 41, p. 107

D(div) domain of the L2-divergence, p. 33D(div) domain of the measure-valued divergence, p. 33

DTV(div) domain of the measure-valued divergence withfinite total variation, p. 33

Test(M ) test functions, p. 41TestL∞(M ) test functions with bounded Laplacian, p. 41

G L1-functions with L1-differential, p. 60Greg closure of G∩ Test(M ) in G, p. 61

D(Hess) domain of the Hessian, p. 48Dreg(Hess) closure of Test(M ) in D(Hess), p. 63

D2,2,1(Hess) F-functions with L1-Hessian, p. 63L2(T ∗M ) cotangent module, p. 27L2(TM ) tangent module, p. 32

L0(T ∗M ) L0-module induced by L2(T ∗M ), p. 44L0(TM ) L0-module induced by L2(TM ), p. 44

Lp(T ∗M ) Lp-space induced by L0(T ∗M ), p. 44Lp(TM ) Lp-space induced by L0(TM ), p. 44

L2((T ∗)⊗2M ) two-fold tensor product of L2(T ∗M ), p. 45L2(T⊗2M ) two-fold tensor product of L2(TM ), p. 45

L2(ΛkT ∗M ) k-fold exterior product of L2(T ∗M ), p. 46L2(T srM ) L2-tensor fields of type (r, s), p. 78

Reg(T ∗M ) regular 1-forms, p. 45Reg(TM ) regular vector fields, p. 45

Test(T ∗M ) test 1-forms, p. 45Test(TM ) test vector fields, p. 45

Reg(ΛkT ∗M ) k-fold exterior product of Reg(T ∗M ), p. 47Reg(T⊗2M ) two-fold tensor product of Reg(T ∗M ), p. 46

Reg(T rsM ) regular (r, s)-tensor fields, p. 80Test(ΛkT ∗M ) k-fold exterior product of Test(T ∗M ), p. 46

Test(T⊗2M ) two-fold tensor product of Test(TM ), p. 46W 1,2(TM ) L2-vector fields with L2-covariant derivative, p. 67H1,2(TM ) closure of Reg(TM ) in W 1,2(TM ), p. 69

W (2,1)(TM ) L2-vector fields with L1-covariant derivative, p. 69D() domain of the Bochner Laplacian, p. 74

W 1,2(T srM ) L2-tensor fields of type (r, s) with L2-covariantderivative, p. 79

D(dk) domain of the differential in L2(ΛkT ∗M ), p. 84D(δk) domain of the codifferential in L2(ΛkT ∗M ), p. 89

W 1,2(ΛkT ∗M ) intersection of D(dk) and D(δk), p. 91H1,2(ΛkT ∗M ) closure of Reg(ΛkT ∗M ) in W 1,2(ΛkT ∗M ), p. 91

D(~∆) domain of the Hodge Laplacian, p. 92H(ΛkT ∗M ) harmonic k-forms, p. 92H1,2♯ (TM ) image of H1,2(T ∗M ) under ♯, p. 97

Part 1. First order differential structure

1. Preliminaries. Dirichlet spaces and module theory

1.1. Notations. We start with basic terminologies used all over the paper.


For the remainder of this article, we make the following topological assumption.Compare with Remark 1.3.

Assumption 1.1. (M , τ) is a topological Lusin space, i.e. a continuous injectiveimage of a Polish space, endowed with a σ-finite Borel measure m on M, accordingto Subsection 1.1.1 below, with full topological support.

The given topology τ is considered as understood and does, up to few exceptions,usually not appear in our subsequent notation.

1.1.1. Measures. All very elementary measure-theoretic terminologies are agreedupon [12, 13, 52]. More specific points are shortly addressed now.

The Borel σ-algebra induced by τ is denoted by B(M ), while its Carathéodorycompletion w.r.t. a Borel measure µ on M is denoted by B

µ(M ). (If not explicitlystated otherwise, we identify certain subsets of M with their equivalence classesin B

µ(M ).) The support of every Borel measure µ on M is defined [78, Sec. V.1]and denoted by sptµ. By M

+f (M ), M+

σ (M ), M±f (M ) and M

±σ (M ), we intend the

spaces of Borel measures on M which are finite, σ-finite, signed and finite, as wellas signed and σ-finite, respectively. Here, σ-finiteness of µ ∈ M

±σ (M ) refers to the

existence of an increasing sequence of open subsets of M on whose elements µ isfinite. The subscripts R, such as in M

+fR(M ), or E, such as in M

+f (M )E, indicate the

respective subclass of (signed) measures which are Radon or do not charge E-polarsets according to Subsection 1.3.2.

Given any µ ∈ M±σ (M ), there exist unique µ+, µ− ∈ M

+σ (M ) such that

µ = µ+ − µ−.

This is the Jordan decomposition of µ into its positive part µ+ and its negative partµ−. At least one of the measures µ+ or µ− is finite — hence µ+ −µ− is well-defined

— while they are both finite if µ ∈ M±f (M ) [52, Thm. 29.B].

The total variation |µ| ∈ M+σ (M ) of µ ∈ M

±σ (M ) is defined by

|µ| := µ+ + µ−.

This gives rise to the total variation norm

‖µ‖TV := |µ|[M ]

of µ. Note that (M±f (M ), ‖ · ‖TV) is indeed a normed vector space.

In this notation, given a not necessarily nonnegative ν ∈ M±σ (M ), we write

ν ≪ µ if ν ≪ |µ|, or equivalently |ν| ≪ |µ| [52, Thm. 30.A], for absolute continuity.Note that if µ, ν ∈ M

+f (M ) are singular to each other, written µ ⊥ ν, then

|µ+ ν| = |µ| + |ν|. (1.1)

Lastly, recall that given any µ, ν ∈ M±σ (M ), µ admits a Lebesgue decomposition

w.r.t. ν [52, Thm. 32.C] — that is, there exist unique µ≪, µ⊥ ∈ M±σ (M ) whose sum

is well-defined and with the property that

µ≪ ≪ ν,

µ⊥ ⊥ ν,

µ = µ≪ + µ⊥.

1.1.2. Functions. Let L0(M ) and L∞(M ) be the spaces of real-valued and boundedreal-valued B(M )-measurable functions defined everywhere on M . Write SF(M )for the space of simple functions, i.e. of those f ∈ L∞(M ) with finite range.

Let µ be a Borel measure on M . Let L0(M , µ) be the real vector space ofequivalence classes of elements in L0(M ) w.r.t. µ-a.e. equality. We mostly makeneither notational nor descriptional distinction between (µ-a.e. properties of) func-tions f ∈ L0(M , µ) and (properties of) its equivalence class [f ]µ ∈ L0(M , µ). In

12 MATHIAS BRAUN

the only case where this difference matters, see Subsection 1.3.2 below, we use thedistinguished notations f or f∼.

The support of f ∈ L0(M , µ) is defined as

sptµ f := spt(|f |µ)

and we briefly write spt for sptm. Let L0c(M , µ) be the class of f ∈ L0(M , µ) such

that sptµ f is compact. We write L0(M ) for L0(M ,m).Given p ∈ [1,∞] we denote the (local) p-th order Lebesgue spaces w.r.t. µ by

Lp(M , µ), with the usual norm ‖ · ‖Lp(M ,µ), and Lploc(M , µ). We always abbreviateLp(M ,m) and Lploc(M ,m) by Lp(M ) and Lploc(M ), respectively.

If (M , d) is a metric space — in which case we always assume that τ coincideswith the topology induced by d — we write Lip(M ) for the space of real-valuedLipschitz functions w.r.t. d. Lipbs(M ) is the class of boundedly supported elementsin Lip(M ), i.e. which vanish identically outside a ball in M .

Lastly, the µ-essential supremum of a family (fi)i∈I in L0(M , µ) with arbitrary,not necessarily countable index set I is the minimal function f ∈ L0(M , µ) suchthat f ≥ fi µ-a.e. for every i ∈ I. It exists and is unique. We write

µ-esssupfi : i ∈ I := f,

µ-essinffi : i ∈ I := −µ-esssup−fi : i ∈ I.while the prefixes “µ-” are dropped from these notations if µ = m.

1.2. Riemannian manifolds with boundary. One family of guiding examplesfor our constructions pursued from Chapter 2 on are Riemannian manifolds, possi-bly noncompact and possibly with boundary. Here we collect basic terminologieson these. See [73, 74, 84, 88] for further reading.

1.2.1. Setting. Unless explicitly stated otherwise, any Riemannian manifold M isunderstood to have topological dimension d ≥ 2 and smooth, i.e. to be locallyhomeomorphic to Rd or Rd−1 × [0,∞) depending on whether ∂M = ∅ or ∂M 6= ∅and with smooth transition functions. Recall that a function f : M → R is smooth[74, Ch. 1] if f x−1 : x(U) → R is smooth in the ordinary Euclidean sense for everychart (U, x) on M — if (U, x) is a boundary chart, i.e. U ∩ ∂M 6= ∅, this meansthat f x−1 has a smooth extension to an open subset of Rd. For simplicity, anyRiemannian manifold is assumed to be connected and (metrically) complete. Set

M := M \ ∂M .

Denote by v ∈ M+σR(M ) the Riemannian volume measure induced by the metric

tensor 〈·, ·〉. Let s ∈ M+σR(∂M ) be the surface measure on ∂M .

If ∂M 6= ∅, then ∂M is a smooth codimension 1 submanifold of M . It naturallybecomes Riemannian when endowed with the pullback metric

〈·, ·〉 := ∗〈·, ·〉under the natural inclusion : ∂M → M . The map induces a natural inclusiond : T∂M → TM

∣∣∂M

which is not surjective. In particular, the vector bundles T∂Mand TM

∣∣∂M

do not coincide. Rather, T∂M is identifiable with the codimension 1subbundle d(T∂M ) of TM

∣∣∂M

.

1.2.2. Sobolev spaces on vector bundles. Denote the space of smooth sections of areal vector bundle F over M (or ∂M ) by Γ(F). (This is a slight abuse of notationsince we also denote carré du champs associated to Dirichlet energies by Γ, seeSubsection 1.3.5. However, it will always be clear from the context which meaningis intended.) With a connection ∇ on F — always chosen to be the Levi-Civitaone if F := TM — one can define Sobolev spaces W k,p(F), k ∈ N0 and p ∈ [1,∞),in various ways, e.g. by completing Γc(F) w.r.t. an appropriate norm w.r.t. v (or


s), or in a weak sense [88, Sec. 1.3]. The “natural” approaches all coincide if Mis compact [88, Thm. 1.3.6], but for noncompact M , without further geometricalrestrictions ambiguities may occur [37]. In our work, it is always either clear fromthe context which definition is intended, or precise meanings are simply irrelevantwhen only the compact setting is considered.

Throughout, it is useful to keep in mind the following trace theorem for compactM [88, Thm. 1.3.7]. If M is noncompact, it only holds true locally [88, p. 39].

Proposition 1.2. For every k ∈ N0 and every p ∈ [1,∞), the natural restrictionmap ·

∣∣∂M

: Γ(F) → Γ(F∣∣∂M

) in the spirit of Subsection 1.2.3 below extends to acontinuous — in fact, compact — map from W k+1,p(F) to W k,p(F

∣∣∂M

).

1.2.3. Normal and tangential components. Denote by n ∈ Γ(TM∣∣∂M

) the outwardpointing unit normal vector field at ∂M . It can be smoothly extended to a (non-relabeled) vector field n on an open neighborhood of ∂M [88, Thm. 1.1.7].

The restriction X∣∣∂M

of a given X ∈ Γ(TM ) to ∂M decomposes into a normalpart X⊥ ∈ Γ(TM

∣∣∂M

) and a tangential part X‖ ∈ Γ(TM∣∣∂M

) defined by

X⊥ := 〈X, n〉 n,

X‖ := X∣∣∂M

−X⊥.(1.2)

Unlike the end of Subsection 1.2.1, under a slight abuse of notation, in a unique wayevery X‖ ∈ Γ(T∂M ) can be identified with a tangential element X ∈ Γ(TM

∣∣∂M

),i.e. X⊥ = 0 [88, p. 16]. In turn, such an X can be smoothly extended to an openneighborhood of ∂M [74, Lem. 8.6]. That is, knowing the restrictions to ∂M of allX ∈ Γ(TM ) with purely tangential boundary components suffices to recover theentire intrinsic covariant structure of ∂M .

Similarly, the tangential part tω ∈ Γ(ΛkT ∗M∣∣∂M

) and the normal part nω ∈Γ(ΛkT ∗M

∣∣∂M

), k ∈ N, of a k-form ω ∈ Γ(ΛkT ∗M ) at ∂M are defined by

tω(X1, . . . , Xk) := ω(X‖1 , . . . , X

‖k),

nω := ω∣∣∂M

− tω(1.3)

for every X1, . . . , Xk ∈ Γ(TM∣∣∂M

). Taking tangential and normal parts of differen-tial forms is dual to each other through the Hodge ⋆-operator [88, Prop. 1.2.6].

1.3. Dirichlet forms. In this section, we summarize various important notions ofDirichlet spaces. This survey is enclosed by two examples in Subsection 1.3.9 wefrequently use for illustrative reasons in the sequel.

For further, more detailed accounts, we refer to the books [14, 29, 43, 78].

1.3.1. Basic definitions. We always fix a symmetric, quasi-regular [78, Def. III.3.1]and strongly local Dirichlet form (E,F) with linear domain F := D(E) which isdense in L2(M ). (In our work, symmetry of E may and will be assumed withoutrestriction, see Remark 2.16.) Here, strong locality [29, Def. 1.3.17] means that forevery f, g ∈ F∩ L0

c(M ) such that f is constant on a neighborhood of spt g,

E(f, g) = 0.

Remark 1.3. As every topological space which carries a quasi-regular Dirichlet form(E,F) is, up to removing an E-polar set, a Lusin space [78, Rem. IV.3.2], ourassumption on M stated at the beginning of Section 1.1 is in fact not restrictive.

Remark 1.4. Strong locality is not strictly necessary to run the construction ofthe cotangent module in Section 2.1. One could more generally assume E to havetrivial killing part in its Beurling–Deny decomposition, see [29, Thm. 4.3.4], [43,Thm. 3.2.1] and [72, Thm. 5.1]. However, if E has nontrivial jump part — which

14 MATHIAS BRAUN

precisely distinguishes it from being strongly local [29, Prop. 4.3.1] — the importantcalculus rules from Proposition 1.12 below typically fail.

The triple (M ,E,m) is called Dirichlet space. If we say that a property holds forE, we usually mean that it is satisfied by the pair (E,F). We abbreviate

Fb := F∩ L∞(M ),

Fc := F∩ L0c(M ),

Fbc := Fb ∩ Fc.L0c

By definition, E is closed [78, Def. I.2.3], i.e. (F, ‖ · ‖F) is complete, where∥∥f

∥∥2

F:=

∥∥f∥∥2

L2(M)+E(f),

and Markovian [78, Def. I.4.5], i.e. minf+, 1 ∈ F as well as

E(minf+, 1

)≤ E(f)

for every f ∈ F. Here, we abbreviate

E(f) := E(f, f)

and we do so analogously for the diagonal values of any other bilinear form encoun-tered in the sequel without further notice.

A densely defined, quadratic form on L2(M ) — for notational convenience, weconcentrate on E — is called closable if for every E-Cauchy sequence (fn)n∈N inD(E) with ‖fn‖L2(M) → 0 as n → ∞, we have E(fn) → 0 as n → ∞. E is closableif and only if it has a closed extension [43, p. 4]. Here, we term a sequence (fn)n∈N

in FE-Cauchy if E(fn − fm) → 0 as n,m → ∞, and E-bounded if

supn∈N

E(fn) < ∞.

1.3.2. Basic properties. For all relevant quasi-notions evolving around the definitionof quasi-regularity, we refer to [29, Ch. 1] or [78, Ch. III]. Here, we solely state thefollowing useful properties [78, Prop. IV.3.3] frequently used in our work. (Here andin the sequel, “E-q.e.” and “E-q.c.” abbreviate E-quasi-everywhere [78, Def. III.2.1]and E-quasi-continuous [78, Def. III.3.2], respectively.)

Proposition 1.5. For every quasi-regular Dirichlet form E, the following hold.

(i) F is a separable Hilbert space w.r.t. ‖ · ‖F.(ii) Every f ∈ F has an E-q.c. m-version f .(iii) If a function f is E-q.c. and is nonnegative m-a.e. on an open set U ⊂ M,

then f is nonnegative E-q.e. on U . In particular, an E-q.c. m-respresentativef of any f ∈ F is E-q.e. unique.

(iv) If f ∈ F∩ L∞(M ), then any E-q.c. m-version f of f obeys

|f | ≤ ‖f‖L∞(M) E-q.e.

1.3.3. Extended domain. We now define the main space around which Chapter 2 isbuilt. In view of Definition 2.6, the point is our goal to speak about L2-differentialsdf of appropriate f ∈ L0(M ) without imposing any integrability assumption on f .

The following definition can be found in [72, p. 690].

Definition 1.6. The extended domain Fe of E is defined to consist of all f ∈ L0(M )for which there exists an E-Cauchy sequence (fn)n∈N of elements of F such thatfn → f pointwise m-a.e. as n → ∞.

Example 1.7. In the metric measure terminology of [4, 45], up to a possible “non-Riemannian structure” of M , see Remark 2.16 below, Fe corresponds to the Sobolevclass S2(M ) [45, Def. 2.1.4], see [4, Thm. 6.2].


We say that a sequence (fn)n∈N converges to f in Fe if fn ∈ F for every n ∈ N,(fn)n∈N is E-Cauchy, and fn → f m-a.e. as n → ∞. The following Proposition 1.8is provided thanks to [72, Prop. 3.1, Prop. 3.2]. Let us also set

Feb := Fe ∩ L∞(M ),

Fec := F∩ L0c(M ),

Febc := Feb ∩ Fec.L0c

Proposition 1.8. The extended domain Fe has the following properties.

(i) E uniquely extends to a (non-relabeled) real-valued bilinear form on F2e in

such a way that for every f ∈ Fe,

E(f) = limn→∞

E(fn)

for every sequence (fn)n∈N that converges to f in Fe.(ii) E

1/2 is a seminorm on Fe.(iii) A function f ∈ L0(M ) belongs to Fe if and only if there exists an E-bounded

sequence (fn)n∈N in Fe such that fn → f m-a.e. as n → ∞. In other words,the functional E1 : L0(M ) → [0,∞] defined by

E1(f) :=

E(f) if f ∈ Fe,

∞ otherwise

is lower semicontinuous w.r.t. pointwise m-a.e. convergence. In particular,E1 viewed as a functional from L2(M ) with values in [0,∞] is convex andL2-lower semicontinuous.

(iv) If f, g ∈ Fe, then minf, g,minf+, 1 ∈ Fe with

E(minf, g

)≤ E(f) +E(g),

E(minf+, 1

)≤ E

(f,minf+, 1

).

(v) We have

F= Fe ∩ L2(M ).

(vi) Every f ∈ Fe has an E-q.c. m-version f which is E-q.e. unique.

In general, unfortunately, the constant function 1M does not belong to Fe. Sincewe nevertheless need approximations to 1M in F at various instances, we recordthe following result due to [72, Thm. 4.1].

Lemma 1.9. There exists a sequence (Gn)n∈N of E-quasi-open Borel subsets of Msuch that Gn ⊂ Gn+1 E-q.e. for every n ∈ N,

⋃n∈N

Gn covers M up to an E-polarset, and for every n ∈ N there exists gn ∈ Fb such that

gn = 1 m-a.e. on Gn.

Remark 1.10. In the terminology of [72], Lemma 1.9 asserts that 1M belongs to thelocal space Floc. It contains Fe [72, Thm. 4.1], but this inclusion may be strict. Forinstance, the energy measure µ from Subsection 1.3.4 below does in general notextend to a bilinear form on F

2

loc in any reasonable sense [33, Rem. 2.13].

Remark 1.11. Unlike the setting of Example 1.19 which is mostly worked upon in[45], Lemma 1.9 does not provide a global control on (E(gn))n∈N. In particular, wedo not know in general whether E(gn) → 0 as n → ∞, a property which is closelyrelated to stochastic completeness of M , see Definition 4.16 and Remark 4.25.

16 MATHIAS BRAUN

1.3.4. Energy measures. E can be represented in terms of so-called energy measures— a fact which also holds if the form admits no killing but a nontrivial jumppart as described in Remark 1.4, see [43, Thm. 3.2.1] and its proof — a conceptwhich is recorded now. This leads to the notion of carré du champ discussed inSubsection 1.3.5. In Section 2.2, it is furthermore used to give a precise meaningto the “differential of a function f ∈ Fe on a Borel set A ⊂ M”.

The class Fb is an algebra w.r.t. pointwise multiplication [14, Prop. I.2.3.2] whichis dense in F [72, Cor. 2.1]. In fact,

E(f g) ≤ 2∥∥f

∥∥2

L∞(M)E(g) + 2

∥∥g∥∥2

L∞(M)E(f)

for every f, g ∈ Fb. Together with [72, Thm. 5.2], there exists a symmetric bilinearmap µ : F2

b → M±fR(M )E such that for every f, g, h ∈ Fb,

2ˆ

M

hdµf,g = E(f h, g) +E(g h, f) −E(f g, h).

The map µ is uniquely determined by this identity. It uniquely extends to a (non-relabeled) symmetric bilinear form µ : F2

e → M±fR(M )E by approximation in Fe [72,

Ch. 5]. The diagonal of µ takes values in M+fR(M )E. Depending on the context,

the map µ or, given any f ∈ Fe, the measure µf are called the energy measure(associated to f).

The following facts about µ with Fe and Feb replaced by Fand Fb, respectively,have been proven in [33, Thm. 2.8] and [72, Lem. 5.2]. By approximation in Fe,any of these properties extend to the former classes. See also Remark 1.15 belowfor slightly more refined statements.

Proposition 1.12. The map µ satisfies the following obstructions.

(i) Representation formula. For every f, g ∈ Fe,

E(f, g) = µf,g[M ].

(ii) Cauchy–Schwarz inequality. For every f, g ∈ Fe and every u, v ∈ L∞(M ),∣∣∣ˆ

M

u v dµf,g

∣∣∣ ≤[ˆ

M

u2 dµf

]1/2 [ˆ

M

v2 dµg

]1/2

.

(iii) Truncation. For every f, g, h ∈ Fe, minf, g ∈ Fe and

µminf,g,h = 1f≤g

µf,h + 1f>g

µg,h,

1f=0

µf = 0.

(iv) Chain rule. For every k, l ∈ N, every f ∈ Fkeb and g ∈ F

leb as well as every

ϕ ∈ C1(Rk) and ψ ∈ C1(Rl) — with ϕ(0) = ψ(0) = 0 if m[M ] = ∞ — wehave ϕ f, ψ g ∈ Fe with

µϕf,ψg =k∑

i=1

l∑

j=1

[∂iϕ f

] [∂jψ g

]µfi,gj

.

(v) Leibniz rule. For every f, g, h ∈ Feb, we have f g ∈ Feb and

µf g,h = f µg,h + gµf,h.

(vi) Strong locality. For every E-quasi-open G ⊂ M, every f ∈ Fe such that fis constant m-a.e. on G and every g ∈ Fe,

1G µf,g = 0.


1.3.5. Carré du champ and E-dominance. Following [33, 60], we now record theconcept of carré du champs and (minimal) E-dominant measures. The secondnotion will especially be relevant in studying the “universality” of the cotangentmodule L2(T ∗M ) and its compatibility to [10, 30, 36, 61, 62, 68], see Theorem 2.9and Remark 2.15, but also in Remark 1.31 below.

Towards the aim of representing E in terms of a “scalar product of gradients”,the following definition from [33, Def. 2.16] and [60, Def. 2.1] is useful.

Definition 1.13. A Borel measure µ on M is called

a. E-dominant if it is σ-finite and µf ≪ µ for every f ∈ F, andb. minimal E-dominant if it is E-dominant and µ ≪ ν for every E-dominant

Borel measure ν on M.

Any two minimal E-dominant measures are mutually equivalent. The class off ∈ F for which µf is minimal E-dominant is dense in F, and every minimalE-dominant measure does not charge E-polar sets [33, Prop. 2.18].

Given any E-dominant µ, there exists a unique symmetric bilinear Γµ : F2e →

L1(M , µ) such that for every f, g ∈ Fe,

E(f, g) =ˆ

M

Γµ(f, g) dµ.

In particular, µf ≪ µ for every f ∈ Fe [60, Lem. 2.2], and the calculus rules fromProposition 1.12 transfer accordingly to Γµ at the µ-a.e. level.

We say that (M ,E,m) or simply E admits a carré du champ if m is E-dominant,in which case we abbreviate Γm by Γ and term it the carré du champ (operator)associated with E. In fact, if E admits a carré du champ, then m is already minimalE-dominant according to the remark after [33, Def. 2.16].

Remark 1.14. In general, E might not always admit a carré du champ [33, Ex. 2.17],which actually motivated Definition 1.13 in [33, 60]. This lack of Γ, however, isexcluded later in Assumption 3.1.

Remark 1.15. Following [14, Thm. I.7.1.1] or [29, Thm. 4.3.8], we have

f♯µf ≪ L1 (1.4)

for every f ∈ Fe. In other words, given an E-dominant µ, Γµ(f) = 0 holds µ-a.e. onf−1(C) for every L

1-negligible Borel set C ⊂ R. In particular, by approximation ofLipschitz by C1-functions, see e.g. the proof of [45, Thm. 2.2.6], the same conclusionas in (iv) in Proposition 1.12 holds for f ∈ F

ke and g ∈ F

le under the hypotheses

that ϕ ∈ Lip(Rk) and ψ ∈ Lip(Rl), where the partial derivatives ∂iϕ and ∂jψ aredefined arbitrarily on their respective sets of non-differentiability points.

If E admits a carré du champ, (1.4) translates into

f♯[Γ(f)m

]≪ L

1.

for every f ∈ Fe. In this framework all E-q.c. representatives in Proposition 1.12can equivalently be replaced by their genuine m-versions. This fact will be used inthe sequel when referring to Proposition 1.12 without further mention.

1.3.6. Neumann Laplacian. By [43, Thm. 1.3.1], E is uniquely associated to a non-positive, self-adjoint — hence closed and densely defined — operator ∆ on L2(M )with domain D(∆) as follows. A function f ∈ F belongs to D(∆) if and only ifthere exists h ∈ L2(M ) such that for every g ∈ F,

−ˆ

M

g h dm = E(g, f). (1.5)

18 MATHIAS BRAUN

If such a function h exists, the element

∆f := h

is unique and termed (Neumann) Laplacian of f . See also Example 1.18 below.If E admits a carré du champ, from Proposition 1.12 the following is straightfor-

ward to derive, see e.g. [14, Sec. I.6].

Lemma 1.16. If E admits a carré du champ, the following hold.

(i) Leibniz rule. For every f, g ∈ D(∆) ∩ L∞(M ) with Γ(f),Γ(g) ∈ L∞(M ),

∆(f g) = f ∆g + 2 Γ(f, g) + g∆f m-a.e.

(ii) Chain rule. For every f ∈ D(∆) ∩ L∞(M ) with Γ(f) ∈ L∞(M ) and everyϕ ∈ C∞(I) for some interval I ⊂ R which contains 0 and the image of f

— with ϕ(0) = 0 if m[M ] = ∞ — we have

∆(ϕ f) =[ϕ′ f

]∆f +

[ϕ′′ f

]Γ(f) m-a.e.

1.3.7. Neumann heat flow. By the spectral theorem [43, Lem. 1.3.2], ∆ induces astrongly continuous semigroup (pt)t≥0 — the so-called (Neumann) heat semigroupor (Neumann) heat flow — of linear operators on L2(M ) by

pt := e∆t.

The curve t 7→ ptf belongs to C1((0,∞); L2(M )) with

ddt

ptf = ∆ptf

for every f ∈ L2(M ) and every t > 0, and for such f , the function t 7→ E(ptf) isnonincreasing on (0,∞). The heat flow commutes with ∆, i.e.

pt ∆ = ∆ pt on D(∆)

for every t ≥ 0. Moreover, if f ∈ F then ptf → f in F as t → 0.For every t ≥ 0, pt is a bounded, self-adjoint operator on L2(M ) with norm

no larger than 1. By the first Beurling–Deny criterion [31, Thm. 1.3.2], (pt)t≥0 ispositivity preserving, i.e. for every nonnegative f ∈ L2(M ), we have ptf ≥ 0 m-a.e.Moreover, by [31, Thm. 1.3.3], (pt)t≥0 is sub-Markovian, i.e. if f ≤ 1M m-a.e. thenptf ≤ 1M m-a.e. as well for every t ≥ 0. Hence, (pt)t≥0 extends to a semigroup ofbounded contraction operators from Lp(M ) to Lp(M ) for every p ∈ [1,∞] which isstrongly continuous if p < ∞ and weakly∗ continuous if p = ∞. Finally, at variousoccasions we will need the subsequent standard a priori estimates, see e.g. [19] orthe arguments in [45, Subsec. 3.4.4].

Lemma 1.17. For every f ∈ L2(M ) and every t > 0,

E(ptf) ≤ 12t

∥∥f∥∥2

L2(M),

∥∥∆ptf∥∥2

L2(M)≤ 1

2t2∥∥f

∥∥2

L2(M).

1.3.8. Associated Markov process. By quasi-regularity [78, Thm. IV.3.5], E is prop-erly associated to an m-reversible Markov process M := (Ω,F, (bt)t≥0, (Px)x∈M†

)consisting of a process (bt)t≥0 with values in a Hausdorff topological space M† :=M ⊔ † with cemetery † and associated lifetime ζ : Ω → [0,∞] defined on a mea-surable space (Ω,F), a family (Px)x∈M†

of probability measures on (Ω,F), andan (implicitly given) filtration (Ft)t≥0 w.r.t. which ((bt)t≥0, (Px)x∈M†

) satisfies theMarkov property. See [78, Sec. IV.1] for precise definitions. The indicated associa-tion is that for every f ∈ L2(M ) and every t > 0,

ptf = E ·[f(b2t) 1t<ζ/2

]m-a.e.,


and the map x 7→ Ex[f(b2t) 1t<ζ/2] defined on M is E-q.c. [78, Def. IV.2.5]. Theseexpressions make sense regardless of the chosen m-version of f [78, Rem. IV.2.6] andrespect the convention f(†) := 0 for f initially defined on M . M can be constructedto be right [78, Def. IV.1.8] and m-tight special standard [78, Def. IV.1.13]. Amongother things, this means that (b0)♯Px = δx for every x ∈ M† and that (bt)t≥0 obeysthe strong Markov property w.r.t. (Ft)t≥0.

By strong locality of E, see e.g. [78, Thm. 1.5] or [29, Thm. 4.3.4], M can beeven chosen to satisfy the a.s. sample path continuity property

P ·[t 7→ bt is continuous on [0, ζ/2)

]= 1 E-q.e.

1.3.9. Two guiding examples. The following frameworks frequently serve as guidingexamples throughout our treatise and are mainly listed to fix notation.

Example 1.18 (Riemannian manifolds with boundary). Let M be a Riemannianmanifold with boundary as in Section 1.2, and let m be a Borel measure on Mwhich is locally equivalent to v. Let W 1,2(M ) be the Sobolev space w.r.t. m

defined in the usual sense on M . Define E: W 1,2(M ) → [0,∞) through

E(f) :=ˆ

M

|∇f |2 dm

and the quantity E(f, g), f, g ∈ W 1,2(M ), by polarization. Then (E,W 1,2(M )) isa Dirichlet form which is strongly local and regular, since C∞

c (M ) is a dense setof W 1,2(M ) which is also uniformly dense in C0(M ). (The latter is the space ofcontinuous functions on M vanishing at ∞.) E admits a carré du champ which isprecisely given by |∇ · |2. See [29, 31, 43, 99] for details.

Furthermore, suppose that m := e−2wv for some w ∈ C2(M ). By Green’s for-

mula, see e.g. [73, p. 44], ∆ is the self-adjoint realization of the drift Laplacian

∆0 − 2 〈∇w,∇·〉

w.r.t. Neumann boundary conditions, where ∆0 is the Laplace–Beltrami operatoron M , initially defined on functions f ∈ C∞

c (M ) with

df(n) = 0 on ∂M .

This equation makes sense s-a.e. for every f ∈ D(∆) by the local trace theorem.

Example 1.19 (Infinitesimally Hilbertian metric measure spaces). Let (M , d,m) bean infinitesimally Hilbertian metric measure space, according to (4.19) in [5], forwhich m satisfies the growth condition (4.2) in [4]. In this case E is the Cheegerenergy introduced in [4, Thm. 4.5], with domain denoted by W 1,2(M ). By [87,Thm. 4.1], E is quasi-regular and strongly local, and it admits a carré du champwhich m-a.e. coincides with the minimal relaxed gradient from [4, Def. 4.2].

Remark 1.20 (Subsets). Let (M , d,m) be as in Example 1.19 and E ⊂ M be a closedsubset. Assume that E = E

, m[E] > 0, m[∂E] = 0, and that the length distance

dE on E2 induced by d is nondegenerate. Then (E, dE ,mE), where mE := m[ · ∩E],induces a quasi-regular, strongly local Dirichlet space (E,EE ,mE) [98, 99] whosecarré du champs coincide mE-a.e. on E. Moreover, in the sense of restrictions offunctions, see e.g. (47) in [98], we have

W 1,2(M )∣∣E

⊂ W 1,2(E). (1.6)

Whenever M fits into Example 1.18 or Example 1.19, we intend the canonicallyinduced Dirichlet space (M ,E,m) without further notice.

20 MATHIAS BRAUN

1.4. L∞-modules. Throughout this section, we assume that µ is a σ-finite Borelmeasure on M . The theory of Lp(M , µ)-normed L∞(M , µ)-modules, p ∈ [1,∞],over general measure spaces has been introduced in [45] and is recorded now.

As in Subsection 1.1.2, when µ = m — which will be the relevent case in mostof our work unless in Chapter 2 — we drop the measure from the notation.

1.4.1. Definition and basic properties.

Definition 1.21. Given p ∈ [1,∞], a real Banach space (M, ‖ · ‖M) or simply M

is termed an Lp-normed L∞-module (over M ) if it comes with

a. a bilinear map · : L∞(M , µ) × M → M satisfying

(f g) · v = f · (g · v),

1M · v = v,

b. a nonnegatively valued map | · |µ : M → Lp(M , µ) such that

|f · v|µ = |f | |v|µ µ-a.e.,

‖v‖M =∥∥|v|µ

∥∥Lp(M ,µ)

,

for every f, g ∈ L∞(M , µ) and every v ∈ M. If only item a. is satisfied, we call(M, ‖ · ‖M) or simply M an L∞-premodule.

We neither express the space M nor the reference measure µ in the terminologyof an Lp-normed L∞-module for brevity. In this section, every module is consideredover the same M and w.r.t. the same µ. Later, it will always either be clear fromthe context or explicitly indicated which M and µ are intended.

Remark 1.22. In [45, Def. 1.2.10], spaces obeying Definition 1.21 are called Lp-normed premodules and are as such a priori more general than Lp-normed L∞-modules. However, these notions coincide for p ∈ [1,∞) [45, Prop. 1.2.12]. Whatonly might be missing in the case p = ∞ is the gluing property [45, Def. 1.2.1,Ex. 1.2.5], whose lack will never occur in our work, hence the minor change ofterminology.

Example 1.23. Lp(M , µ) is an Lp-normed L∞-module w.r.t. µ, p ∈ [1,∞].

We call M an L∞-module if it is Lp-normed for some p ∈ [1,∞] — which isassumed throughout the rest of this section — and separable if it is a separableBanach space. We term v ∈ M (µ-essentially) bounded if |v|µ ∈ L∞(M , µ). If M isseparable, it admits a countable dense subset of bounded elements. We drop the ·sign if the multiplication on M is understood. By [45, Prop. 1.2.12], | · |µ is localin the sense that |v|µ = 0 µ-a.e. on E if and only if 1E v = 0 for every v ∈ M andevery E ∈ B

µ(M ). We write

v = 0 := |v|µ = 0,v 6= 0 := v = 0c

Lastly, for every v, w ∈ M we have the µ-a.e. triangle inequality

|v + w|µ ≤ |v|µ + |w|µ µ-a.e.,

which shows that the map | · |µ : M → Lp(M , µ) is continuous.M is called Hilbert module if it is an L2-normed L∞-module and a Hilbert space

[45, Def. 1.2.20, Prop. 1.2.21]. Its pointwise norm | · |µ satisfies a pointwise µ-a.e. parallelogram identity. In particular, it induces a pointwise scalar product〈·, ·〉µ : M2 → L1(M , µ) which is L∞(M , µ)-bilinear, µ-a.e. nonnegative definite,local in both components, satisfies the pointwise µ-a.e. Cauchy–Schwarz inequality,and reproduces the Hilbertian scalar product on M by integration w.r.t. µ.


Let M and N be Lp-normed L∞-modules, p ∈ [1,∞], such that N is a closedsubspace of M . Then the quotient M/N is an Lp-normed L∞-module as well [45,Prop. 1.2.14] with pointwise norm given by

|[v]|µ := µ-essinf

|v + w|µ : w ∈ N.

For instance, given E ∈ Bµ(M ), the L∞-module M

∣∣E

consisting of all v ∈ M suchthat v 6= 0 ⊂ E can be canonically identified with M/M

∣∣Ec .

1.4.2. Duality. Let M and N be L∞-normed modules. Slightly abusing notation,denote both pointwise norms by | · |µ. A map T : M → Nis called module morphismif it is a bounded linear map in the sense of functional analysis and

T (f v) = f T (v)

for every v ∈ M and every f ∈ L∞(M , µ). The set of all such module morphismsis written Hom(M;N) and is equipped with the usual operator norm ‖ · ‖M;N. Weterm M and N isomorphic (as L∞-modules) if there exist T ∈ Hom(M;N) andS ∈ Hom(N;M) such that T S = IdN and S T = IdM. Any such T is calledmodule isomorphism. If in addition, such a T is a norm isometry, it is called moduleisometric isomorphism, while it is a pointwise module isometric isomorphism if iteven preserves pointwise norms µ-a.e., i.e. for every v ∈ M,

|T (v)|µ = |v|µ µ-a.e.

The dual module to M is defined by

M∗ := Hom(M; L1(M , µ))

and will be endowed with the usual operator norm. The pointwise pairing betweenv ∈ M and L ∈ M

∗ is denoted by L(v) ∈ L1(M , µ). If M is Lp-normed, thenM

∗ is an Lq-normed L∞-normed module, where p, q ∈ [1,∞] with 1/p + 1/q = 1[45, Prop. 1.2.14] with naturally defined multiplication and, by a slight abuse ofnotation, pointwise norm given by

|L|µ := µ-esssup

|L(v)| : v ∈ M, |v|µ ≤ 1 µ-a.e.. (1.7)

By [45, Cor. 1.2.16], if p < ∞,

|v|µ = µ-esssup

|L(v)| : L ∈ M∗, |L|µ ≤ 1 µ-a.e.

for every v ∈ M. Moreover, if p < ∞, in the sense of functional analysis M∗ andthe Banach space dual M

′ of M are isometrically isomorphic [45, Prop. 1.2.13].In this case, the natural pointwise pairing map J: M → M∗∗, where M∗∗ :=Hom(M∗; L1(M , µ)), belongs to Hom(M;M∗∗) and constitutes a norm isometry[45, Prop. 1.2.15]. We term M reflexive (as L∞-module) if J is surjective. If M

is Lp-normed for p ∈ (1,∞), this is equivalent to M being reflexive as Banachspace [45, Cor. 1.2.18], while for p = 1, the implication from “reflexive as Banachspace” to “reflexive as L∞-module” still holds [45, Prop. 1.2.13, Prop. 1.2.17]. Inparticular, all Hilbert modules are reflexive in both senses.

If M is a Hilbert module, we have the following analogue of the Riesz represen-tation theorem [45, Thm. 1.2.24]. For v ∈ M, let Lv ∈ M∗ be given by

Lv(w) := 〈v, w〉.Proposition 1.24. Let M be a Hilbert module. Then the map which sends v ∈ M

to Lv ∈ M∗ is a pointwise module isometric isomorphism, and in particular a norm

isometry. Moreover, for every l ∈ M′ there exists a unique v ∈ M with

l =ˆ

M

〈v, ·〉µ dµ.

22 MATHIAS BRAUN

1.4.3. L0-modules. Let M be an L∞-module. Following [45, Sec. 1.3] we now recalla natural concept of building a topological vector space M0 of “measurable elementsof M without integrability restrictions” containing M with continuous inclusion aswell as a (non-relabeled) extension of the pointwise norm | · |µ : M0 → L0(M , µ)such that for every v ∈ M

0, v ∈ M if and only if |v|µ ∈ Lp(M , µ).Let (Bi)i∈N a Borel partition of M such that m[Bi] ∈ (0,∞) for every i ∈ N.

Denote by M0 the completion of M w.r.t. the distance dM0 : M2 → [0,∞) with

dM0 (v, w) :=∑

i∈N

2−i

m[Bi]

ˆ

Bi

min|v − w|, 1 dµ.

We refer to M0 as the L0-module associated to M. The induced topology on M0

does not depend on the choice of (Bi)i∈N [45, p. 31]. Additionally, scalar andfunctional multiplication, and the pointwise norm | · |µ extend continuously to M0,so that all µ-a.e. properties mentioned in Section 1.4 hold for general elements inM0 and L0(M , µ) in place of M and L∞(M , µ). The pointwise pairing of M and M∗

extends uniquely and continuously to a bilinear map on M0 × (M∗)0 with values inL0(M , µ) such that for every v ∈ M0 and every L ∈ (M∗)0,

|L(v)| ≤ |L|µ |v|µ µ-a.e.,

and we have the following characterization of elements in (M∗)0 [45, Prop. 1.3.2].

Proposition 1.25. Let T : M0 → L0(M , µ) be a linear map for which there existsf ∈ L0(M , µ) such that for every v ∈ M,

|T (v)| ≤ f |v|µ µ-a.e.

Then there exists a unique L ∈ (M∗)0 such that for every v ∈ M,

L(v) = T (v) µ-a.e.,

and we furthermore have

|L|µ ≤ f µ-a.e.

Remark 1.26. To some extent, one can make sense of L0-normed modules w.r.t. asubmodular outer measure µ∗ on M [32, Def. 2.4]. A prominent example of sucha µ∗ is the E-capacity capE [32, Def. 2.6, Prop. 2.8], which — towards the aim ofdefining quasi-continuity of vector fields over metric measure spaces — motivatedthe authors of [32] to study this kind of modules. However, what lacks for such µ∗

is a working definition of dual modules, and in particular Proposition 1.25 seemsunavailable [32, Rem. 3.3].

Remark 1.27. The concept of L0-modules is tightly linked to the one of measurablefields of Hilbert spaces [36]. See [45, Rem. 1.4.12] and [61, Ch. 2] for details.

1.4.4. Local dimension and dimensional decomposition. Given an L∞-module M

and E ∈ Bµ(M ), we say that v1, . . . , vn ∈ M, n ∈ N, are independent (on E) if all

functions f1, . . . , fn ∈ L∞(M , µ) obeying

f1 v1 + · · · + fn vn = 0 µ-a.e. on E

vanish µ-a.e. on E. This notion of local independence is well-behaved under passageto subsets and under module isomorphisms, see [45, p. 34] for details. The spanspanE V of a subset V ⊂ M on E ⊂ M — briefly span V if E = M — is thespace consisting of all v ∈ M

∣∣E

which possess the following property: there existsa disjoint partition (Ek)k∈N of E in B

µ(M ) such that for every k ∈ N, we can findmk ∈ N, vk1 , . . . , v

kmk

∈ Vand fk1 , . . . , fkmk

∈ L∞(M , µ) such that

1Ekv = fk1 v

k1 + · · · + fkmk

vkmk.


Its closure cl‖·‖MspanE V is usually referred to as the space generated by Von E,

or simply by Vif E = M [45, Def. 1.4.2].If Vis a finite set, spanE Vis closed [45, Prop. 1.4.6], a fact which gives additional

strength to the following notions [45, Def. 1.4.3].

Definition 1.28. A family v1, . . . , vn ⊂ M, n ∈ N, is a said to be a (local) basis(on E) if v1, . . . , vn are independent on E, and

spanE v1, . . . , vn = M∣∣E.

We say that M has (local) dimension n (on E) if there exists a local basis of M onE. We say that M has infinite dimension on E if it does not have finite dimensionon E according to the previous sentence.

For the well-posedness of this definition and its link to local independence, werefer to [45, Prop. 1.4.4]. If M is Lp-normed w.r.t. µ, p < ∞, and has dimensionn ∈ N on E, then M∗ has dimension n on E as well [45, Thm. 1.4.7.].

The following important structural result is due to [45, Prop. 1.4.5].

Proposition 1.29. For every L∞-module M w.r.t. µ, there exists a unique Borelpartition (En)n∈N∪∞ of M such that

a. for every n ∈ N with µ[En] > 0, M has dimension n on En, andb. for every E ∈ B

µ(M ) with E ⊂ E∞, M has infinite dimension on E.

Remark 1.30. Using Proposition 1.29, it is possible to establish a one-to-one-cor-respondence between separable Hilbert modules and direct integrals of separableHilbert spaces [45, Thm. 1.4.11, Rem. 1.4.12]. Albeit at a structural level, thisprovides the link of Chapter 2 and Chapter 3 to earlier axiomatizations of spacesof 1-forms and vector fields on Dirichlet spaces [10, 30, 36, 61, 62, 68], and in viewof the universal property of Theorem 2.9 below, we do not enter into details hereand leave these to the interested reader. We only point out the remark at [45, p. 42]that the interpretation of this link should be treated with some care.

Remark 1.31 (Hino index). Unlike Proposition 1.29, in Dirichlet form theory therealready exists a natural notion of “pointwise tangent space dimension” in terms ofthe (pointwise) Hino index introduced in [60], which is quickly recorded now. Forthe modules under our consideration, these two notions of local dimension turn outto coincide, see Corollary 2.8 and also Section 5.3.

Let µ be minimal E-dominant. Let (fi)i∈N be a sequence in Fwhose linear spanis dense in F. The pointwise index of (E,F) or simply E [60, Def. 2.9, Prop. 2.10]is the function p: M → N0 ∪ ∞ given by

p := supn∈N

rank[Γµ(fi, fj)

]i,j∈1,...,n

.

See [60, Ch. 2] for a thorough discussion on the well-definedness of p. In particular,by [60, Prop. 2.11] we know that p > 0 µ-a.e. unless E is trivial. The indexp∗ ∈ N ∪ ∞ of (E,F) or simply E is then

p∗ := ‖p‖L∞(M ,µ).

These definitions are independent of the choice of minimal E-dominant µ. By [60,Prop. 2.10], we have the following result. (Another probabilistic aspect of it nottreated in our work is that p∗ coincides with the so-called martingale dimension[59, 71] w.r.t. the Markov process on M from Subsection 1.3.8 [60, Thm. 3.4].)

Lemma 1.32. For every n ∈ N and every f1, . . . , fn ∈ F,

rank[Γµ(fi, fj)

]i,j∈1,...,n

≤ p ≤ p∗ µ-a.e.

Moreover, p is the µ-a.e. smallest function satisfying the first µ-a.e. inequality forevery n ∈ N and every f1, . . . , fn ∈ F.

24 MATHIAS BRAUN

1.4.5. Tensor products. Let M1 and M2 be two Hilbert modules. Again, by a slightabuse of notation, we denote both pointwise scalar products by 〈·, ·〉µ.

Let M01 ⊙M0

2 be the “tensor product” consisting of all finite linear combinationsof formal elements v⊗w, v ∈ M

01 and w ∈ M

02, obtained by factorizing appropriate

vector spaces [45, Sec. 1.5]. It naturally comes with a multiplication · : L0(M , µ) ×(M0

1 ⊙ M02) → L0(M , µ) defined through

f (v ⊗ w) := (f v) ⊗ w = v ⊗ (f w)

and a pointwise scalar product :µ : (M01 ⊙ M

02)2 → L0(M , µ) given by

(v1 ⊗ w1) :µ (v2 ⊗ w2) := 〈v1, v2〉µ 〈w1, w2〉µ, (1.8)

both extended to M01 ⊙M

02 by (bi-)linearity. Then :µ is bilinear, µ-a.e. nonnegative

definite, symmetric, and local in both components [46, Lem. 3.2.19].The pointwise Hilbert–Schmidt norm | · |HS,µ : M0

1 ⊙ M02 → L0(M , µ) is given by

|A|HS,µ :=√A :µ A. (1.9)

This map satisfies the µ-a.e. triangle inequality and is 1-homogeneous w.r.t. multi-plication with L0(M , µ)-functions [45, p. 44].

Consequently, the map ‖ · ‖M1⊗M2 : M01 ⊙ M0

2 → [0,∞] defined through

‖A‖M1⊗M2 :=∥∥|A|HS,µ

∥∥L2(M ,µ)

has all properties of a norm except that it might take the value ∞.

Definition 1.33. The tensor product M1 ⊗M2 is the completion w.r.t. ‖ · ‖M1⊗M2

of the subspace that consists of all A ∈ M01 ⊙ M0

2 such that ‖A‖M1⊗M2 < ∞.

Inductively, for M := M1 and k ∈ N, up to unique identification we set

M⊗k := M

⊗(k−1) ⊗ M = M⊗ M⊗(k−1),

where we conventionally set M⊗0 := L2(M ) as well.Through (1.8), M1 ⊗ M2 naturally becomes a Hilbert module [45, p. 45]. If M1

and M2 are separable, then so is M1 ⊗M2. Indeed, if Di ⊂ Mi are countable densesubsets consisting of bounded elements, i ∈ 1, 2, then the linear span of elementsof the form v ⊗ w, v ∈ D1 and w ∈ D2, is dense in M1 ⊗ M2. Here, boundednessis essential as underlined by the next remark.

Remark 1.34. The space M1 ⊗M2 should not be confused with the tensor productM1 ⊗H M2 in the Hilbert space sense [69]. Indeed, in general these do not coincide[45, Rem. 1.5.2]. For instance, for v ∈ M1 and w ∈ M2 we always have v ⊗H w ∈M1 ⊗H M2 since the corresponding norm is

‖v ⊗H w‖M1⊗HM2 = ‖v‖M1‖w‖M2

,

but according to (1.8), the norm

‖v ⊗ w‖M1⊗M2 =[ˆ

M

|v|2µ |w|2µ dµ]1/2

in M01 ⊙ M

02 might well be infinite unless, for instance, v or w is bounded.

More intuitively, we should think about v ⊗ w ∈ M01 ⊙ M0

2 as section x 7→v(x) ⊗ w(x) over M , while v ⊗H w ∈ M1 ⊗H M2 is interpreted as section (x, y) 7→v(x) ⊗ w(y) over M 2. The latter point of view is not relevant in this work, but iscrucial e.g. in obtaining a spectral representation of the heat kernel on 1-forms oncompact RCD∗(K,N) spaces [16, Thm. 6.11], K ∈ R and N ∈ [1,∞).


Lastly, we introduce the concept of symmetric and antisymmetric parts in thecase M := M1 = M2. Denote by A⊤ ∈ M⊗2 the transpose of A ∈ M⊗2 as definedin [45, Sec. 1.5]. For instance, for bounded v, w ∈ M we have

(v ⊗ w)⊤ = w ⊗ v. (1.10)

It is an involutive pointwise module isometric isomorphism. We shall call A ∈ M⊗2

symmetric if A = A⊤ and antisymmetric if A = −A⊤. We write M⊗2sym and M⊗2

asym

for the subspaces of symmetric and antisymmetric elements in M⊗2, respectively.

These are closed and pointwise µ-a.e. orthogonal w.r.t. :µ. As usual, for everyA ∈ M⊗2 there exist a unique Asym ∈ M⊗2

sym, the symmetric part of A, and aunique Aasym ∈ M⊗2

asym, the antisymmetric part of A, such that

A = Asym +Aasym.

In particular, we have

∣∣A∣∣2

HS,µ=

∣∣Asym

∣∣2

HS,µ+

∣∣Aasym

∣∣2

HS,µµ-a.e. (1.11)

Next, we present a duality formula for symmetric parts crucially exploited later inLemma 8.2. If D ⊂ M is a set of bounded elements generating M in the sense ofSubsection 1.4.4, then v ⊗ v : v ∈ D generates M⊗2

sym, and after [45, Prop. 1.4.9]for every A ∈ M

⊗2 we have the duality formula

∣∣Asym

∣∣2

HS,µ= µ-esssup

2A :

m∑

j=1

vj ⊗ vj −∣∣∣m∑

j=1

vj ⊗ vj

∣∣∣2

HS,µ:

m ∈ N, v1, . . . , vm ∈ D.

(1.12)

1.4.6. Traces. Let M be a Hilbert module over M . In terms of local bases outlinedin Subsection 1.4.4, it is possible to define the trace of an element A ∈ M

⊗2sym.

As usual, Gram–Schmidt orthonormalization combined with [45, Thm. 1.4.11]entails the following. Denoting by (En)n∈N∪∞ the dimensional decomposition ofM according to Proposition 1.29, for every n ∈ N and every Borelian E ⊂ En withµ[E] ∈ (0,∞), there exists a basis en1 , . . . , enn ⊂ M

∣∣E

of M on E with⟨eni , e

nj

⟩µ

= δij µ-a.e. on E (1.13)

for every i, j ∈ 1, . . . , n. Moreover, for every Borelian E ⊂ E∞ with µ[E] ∈ (0,∞)there exists a sequence (e∞

i )i∈N in M∣∣E

which generates M on E and satisfies (1.13)for every i, j ∈ N and n := ∞. Any such en1 , . . . , enn, n ∈ N, or (e∞

i )i∈N is calleda pointwise orthonormal basis of M on E. In particular, for every E ⊂ En as above,n ∈ N ∪ ∞ and every v ∈ M we can write

1E v =n∑

i=1

⟨v, eni

⟩µeni .

Lastly, if E ⊂ En is such a Borel set, n ∈ N ∪ ∞, we define

1E trA :=n∑

i=1

A : (eni ⊗ eni ). (1.14)

This does not depend on the choice of the pointwise orthonormal basis.

26 MATHIAS BRAUN

1.4.7. Exterior products. Let M be a Hilbert module and k ∈ N0. Set Λ0M0 :=

L0(M , µ) and, for k ≥ 1, let ΛkM0 be the “exterior product” constructed by suitablyfactorizing (M0)⊙k [45, Sec. 1.5]. The representative of v1⊙· · ·⊙vk, v1, . . . , vk ∈ M0,in ΛkM0 is written v1 ∧ · · · ∧ vk. ΛkM0 naturally comes with a multiplication· : L0(M , µ) × ΛkM0 → ΛkM0 via

f (v1 ∧ . . . vk) := (f v1) ∧ · · · ∧ vk = · · · = v1 ∧ · · · ∧ (f vk)

and a pointwise scalar product 〈·, ·〉µ : (ΛkM0)2 → L0(M , µ) defined by

〈v1 ∧ · · · ∧ vk, w1 ∧ · · · ∧wk〉µ := det[〈vi, wj〉µ

]i,j∈1,...,k

(1.15)

up to a factor k!, both extended to ΛkM0 by (bi-)linearity. Then 〈·, ·〉µ is bilinear,µ-a.e. nonnegative definite, symmetric, and local in both components.

Remark 1.35. Given any k, k′ ∈ N0, the map assigning to v1 ∧ · · · ∧ vk ∈ ΛkM0

and w1 ∧ · · · ∧ wk′ ∈ Λk′

M0 the element v1 ∧ · · · ∧ vk ∧ w1 ∧ · · · ∧ wk′ ∈ Λk+k′

M0

can and will be uniquely extended by bilinearity and continuity to a bilinear map∧ : ΛkM0 ×Λk

′

M0 → Λk+k′

M0 termed wedge product [45, p. 47]. If k = 0 or k′ = 0,

it simply corresponds to multiplication of elements of ΛkM0 or Λk′

M0, respectively,with functions in L0(M , µ) according to (1.15).

By a slight abuse of notation, define the map | · |µ : ΛkM0 → L0(M , µ) by

|ω|µ :=√

〈ω, ω〉µ.It obeys the µ-a.e. triangle inequality and is homogeneous w.r.t. multiplication withL0(M , µ)-functions [45, p. 47].

It follows that the map ‖ · ‖ΛkM : ΛkM0 → [0,∞] defined by

‖ω‖ΛkM :=∥∥|ω|µ

∥∥L2(M ,µ)

has all properties of a norm except that ‖ω‖ΛkM might be infinite.

Definition 1.36. The (k-fold) exterior product ΛkM is defined as the completionw.r.t. ‖ · ‖ΛkM of the subspace consisting of all ω ∈ ΛkM0 such that ‖ω‖ΛkM < ∞.

The space ΛkM naturally becomes a Hilbert module and, if M is separable, isseparable as well [45, p. 47].

2. Cotangent module

In this chapter, following [45, Sec. 2.2] we discuss a key object of our treatise,namely the cotangent module L2(T ∗M ), i.e. the space of differential 1-forms thatare square-integrable in a certain “universal” sense made precise in Theorem 2.4.

2.1. The construction. Define the pre-cotangent module Pcm by

Pcm :=

(fi, Ai)i∈N : (Ai)i∈N partition of M in B(M ),

(fi)i∈N in Fe,∑

i∈N

µfi[Ai] < ∞

.

Moreover, define a relation R on Pcm by declaring that (fi, Ai)i∈N R (gj , Bj)j∈N

if and only if µfi−gj[Ai ∩Bj ] = 0 for every i, j ∈ N. R is in fact an equivalence

relation by Proposition 1.12. The equivalence class of an element (fi, Ai)i∈N ∈Pcm w.r.t. R is shortly denoted by [fi, Ai]. As made precise in Theorem 2.4 andDefinition 2.6, we think of [fi, Ai] ∈ Pcm/R as the 1-form which equals dfi on Aifor every i ∈ N in a certain “universal” a.e. sense.

Pcm/R becomes a vector space via the well-defined operations

[fi, Ai] + [gj , Bj] := [fi + gj , Ai ∩Bj ],


λ [fi, Ai] := [λ fi, Ai]

for every [fi, Ai], [gj , Bj ] ∈ Pcm/R and every λ ∈ R.In an analogous way, we define an action of SF(M ) on Pcm/R as follows. Let

h ∈ SF(M ) and [fi, Ai] ∈ Pcm/R. Write

h = 1B1 h1 + · · · + 1Bkhk (2.1)

where k ∈ N, B1, . . . , Bk ∈ B(M ) are disjoint, and h1, . . . , hk ∈ R. Define

h [fi, Ai] := [hj fi, Ai ∩Bj], (2.2)

where we set Bj := ∅ and hj := 0 for every integer j > k. It is straightforwardto verify that this definition is well-defined, independent of the particular way ofwriting h, and gives rise to a bilinear map SF(M ) × Pcm/R → Pcm/R such thatfor every [fi, Ai] ∈ Pcm/R and every h, k ∈ SF(M ),

(h k) [fi, Ai] = h(k [fi, Ai]

),

1M [fi, Ai] = [fi, Ai].

Lastly, by Proposition 1.12, the map ‖ · ‖L2(T∗M) : Pcm/R → [0,∞) given by∥∥[fi, Ai]

∥∥2

L2(T∗M):=

∑

i∈N

µfi[Ai]

constitutes a norm on Pcm/R.

Definition 2.1. We define the Banach space (L2(T ∗M ), ‖ · ‖L2(T∗M)) as the com-pletion of (Pcm/R, ‖ · ‖L2(T∗M)). The pair (L2(T ∗M ), ‖ · ‖L2(T∗M)) or simply L2(T ∗M )is henceforth called cotangent module, and the elements of L2(T ∗M ) are calledcotangent vector fields or (differential) 1-forms.

Remark 2.2. The name “cotangent module” for L2(T ∗M ) is justified by Theorem 2.4below. We point out for now that the notation L2(T ∗M ) is purely formal since wedid and do not define any kind of cotangent bundle T ∗M . It rather originates in theanalogy of L2(T ∗M ) with the space of L2-sections of the cotangent bundle T ∗Min the smooth setting described in Remark 2.13 below. On the other hand, by thestructural characterization of Hilbert modules as direct integral of measurable fieldsof certain Hilbert spaces (Hx)x∈M , see e.g. [61, p. 4381] and Remark 2.15 below,one could think of a fictive cotangent bundle as something “a.e. defined”. This pointof view has been taken in the approaches [10, 30, 36, 61, 62, 68].

A main ingredient to establish Theorem 2.4 is the following lemma, which isan immediate consequence of the above construction and the density of SF(M ) inL∞(M ) w.r.t. the uniform norm.

Lemma 2.3. The map from SF(M )×Pcm/R into Pcm/R defined in (2.2) extendscontinuously and uniquely to a bilinear map from L∞(M ) × L2(T ∗M ) into L2(T ∗M )satisfying, for every f, g ∈ L∞(M ) and every ω ∈ L2(T ∗M ),

(f g)ω = f (g ω),

1M ω = ω,

‖f ω‖L2(T∗M) ≤ sup |f |(M ) ‖ω‖L2(T∗M).

Theorem 2.4 (Module property). For every E-dominant Borel measure µ on M,the cotangent module L2(T ∗M ) is an L2-normed L∞-module over M w.r.t. µ whosepointwise norm | · |µ satisfies, for every [fi, Ai] ∈ Pcm/R,

∣∣[fi, Ai]∣∣µ

=∑

i∈N

1AiΓµ(fi)1/2 µ-a.e. (2.3)

In particular, L2(T ∗M ) is a Hilbert module w.r.t. µ.

28 MATHIAS BRAUN

Proof. ByE-dominance, passing to µ-versions in L∞(M ) of any given f ∈ L∞(M , µ)in the action of L∞(M ) on L2(T ∗M ) induces a well-defined bilinear map fromL∞(M , µ) × L2(T ∗M ) to L2(T ∗M ) which, thanks to Lemma 2.3, turns L2(T ∗M )into an L∞-premodule w.r.t. µ.

Now define the pointwise norm | · |µ : Pcm/R → L2(M , µ) by (2.3). This map isclearly an isometry from Pcm/R to L2(M , µ), whence by continuous and uniqueextension to L2(T ∗M ), it will only be necessary to prove the required properties for| · |µ from Definition 1.21 for elements of Pcm/R and SF(M ), respectively. Indeed,for [fi, Ai] ∈ Pcm/R, by Fubini’s theorem,

ˆ

M

∣∣[fi, Ai]∣∣2

µdµ =

∑

i∈N

ˆ

Ai

Γµ(fi) dµ =∑

i∈N

µfi[Ai] =

∥∥[fi, Ai]∥∥2

L2(T∗M).

On the other hand, writing a given h ∈ SF(M ) according to (2.1),∣∣h [fi, Ai]

∣∣µ

=∑

i,j∈N

1Ai1Bj

|hj| Γµ(fi)1/2 = |h|∣∣[fi, Ai]

∣∣µ

µ-a.e.,

which establishes the desired properties for | · |µ.The last statement follows since ‖ · ‖L2(T∗M) satisfies the parallelogram identity,

which is a consequence of the bilinearity and symmetry of µ.

Remark 2.5. Conceptually, one could alternatively construct L2(T ∗M ) as follows.Given an E-dominant µ, define the “L0-module” — put in quotes since it is not apriori induced by an L∞-module — L0(T ∗M )µ w.r.t. µ as completion w.r.t. an ap-propriate distance constructed from Γ1/2

µ , compare with Subsection 1.4.3. Thenrestrict to the subspace of elements whose induced pointwise norm belongs toL2(M , µ). See [36, 46, 61] for details. The advantage of our above approach isthat it is clearer in advance that the resulting space does not depend on µ.

2.2. Differential of a function in the extended domain. L2(T ∗M ) directlyprovides a notion of a differential acting on functions in Fe. The behavior of a givenelement of L2(T ∗M ) is completely determined by its interaction with differentials offunctions in Fe, see Theorem 2.9. In turn, this will be used to phrase the calculusrules from Proposition 1.12 at the L∞-module level, see Proposition 2.11.

We start with the following definition.

Definition 2.6. The differential of any function f ∈ Fe is defined by

df := [f,M ],

where [f,M ] ∈ Pcm/R is the representative of the sequence (fi, Ai)i∈N given byf1 := f , A1 := M, fi := 0 and Ai := ∅ for every i ≥ 2.

As usual, we call a 1-form ω ∈ L2(T ∗M ) exact if, for some f ∈ Fe,

ω = df.

The differential d is a linear operator on Fe. By (2.3), w.r.t. the L∞-modulestructure induced by any E-dominant µ according to Theorem 2.4,

|df |µ = Γµ(f)1/2 µ-a.e. (2.4)

holds for every f ∈ Fe.

2.2.1. Universality. To derive the calculus rules from Proposition 2.11 relying onProposition 1.12, we need to prove the following density property. It is independentof the particular choice of the E-dominant Borel measure µ on M that inducesthe background L∞-module structure on L2(T ∗M ) according to Theorem 2.4. Arelated version involving the local density of “regular vector fields” which relies onthe second order calculus developed in Part 2 is stated in Lemma 8.20 below.


Lemma 2.7. The cotangent module L2(T ∗M ) is generated, in the sense of L∞-modules, by dFe and by dF. In particular, L2(T ∗M ) is separable.

Proof. By the definition of the norm ‖ · ‖L2(T∗M) on Pcm/R and after passing tothe limit, we see that the family S of finite linear combinations of objects 1A df ,A ∈ B(M ) and f ∈ Fe, is dense in Pcm/R. Given that Pcm/R is dense in L2(T ∗M )by construction of the latter space, the first claim follows.

Next, given any A ∈ B(M ) and f ∈ Fe, by definition of Fe and Proposition 1.12there exists a sequence (fn)n∈N in F such that

limn→∞

∥∥1A d(fn − f)∥∥2

L2(T∗M)= limn→∞

µfn−f [A] ≤ limn→∞

E(fn − f) = 0.

Thus, dF generates S, and by the argument from the first part of the proof and adiagonal procedure, dF generates L2(T ∗M ).

To see the separability of L2(T ∗M ), note that dF is a separable subset since‖df‖L2(T∗M) ≤ ‖f‖F for every f ∈ F and by the separability of F granted byProposition 1.8. In particular, L2(T ∗M ) is separable by [45, Prop. 1.4.10].

Corollary 2.8. Let (En)n∈N∪∞ be the dimensional decomposition of L2(T ∗M ),seen as an L2-normed L∞-module w.r.t. a given E-dominant µ. Denote by p thepointwise index from Remark 1.31. Then for every n ∈ N ∪ ∞,

p = n µ-a.e. on En.

Theorem 2.9 (Universal property). Let µ be an E-dominant Borel measure on M.Let (M,d) be a tuple consisting of an L2-normed L∞-module over M w.r.t. µ withpointwise norm denoted by µ| · | and a linear map d: Fe → M such that

a. M is generated by dFe, andb. for every f ∈ Fe,

µ|df | = Γµ(f)1/2 µ-a.e.

Then there exists a unique module isomorphism Φ: L2(T ∗M ) → M such that

Φ d = d. (2.5)

Proof. First, we observe that for every f, g ∈ Fe and every Borel subset A ⊂ M , wehave 1A df = 1A dg w.r.t. the L∞-module structure of L2(T ∗M ) from Theorem 2.4if and only if 1Adf = 1Adg w.r.t. the given L∞-module structure of M (bothw.r.t. µ). Indeed, this follows from combining Theorem 2.4 with b.:

1A∣∣d(f − g)

∣∣µ

= 1A Γµ(f − g)1/2 = 1A∣∣µ

∣∣d(f − g)∣∣ µ-a.e. (2.6)

Define the real vector space Q ⊂ M by

Q := ∑

i∈N

1Aidfi : (Ai)i∈N partition of M in B(M ),

(fi)i∈N in Fe,∑

i∈N

∥∥1Ai µ|dfi|∥∥2

L2(M ,µ)< ∞

,

as well as the map Φ: Pcm → Q by

Φ∑

i∈N

1Aidfi :=

∑

i∈N

1Aidfi. (2.7)

By (2.6), Φ is well-defined, it is linear, and by definition (2.5) holds. Moreover, therelation (2.7) will entail the claimed uniqueness of the continuous extension of Φ toL2(T ∗M ) as a byproduct. To prove the actual existence of such an extension, wefirst observe that by (2.6),∣∣∣

µ

∣∣∣∑

i∈N

1Aidfi

∣∣∣ =∣∣∣∑

i∈N

1Aidfi

∣∣∣µ

µ-a.e.

30 MATHIAS BRAUN

In particular, Φ is a norm isometry, i.e.∥∥Φω

∥∥M

= ‖ω‖L2(T∗M)

for every ω ∈ Pcm. By the density of Pcm/R in L2(T ∗M ), Φ extends uniquelyand continuously to a linear isometry Φ: L2(T ∗M ) → M which also preserves therespective pointwise norms. The latter map is injective, has closed image and isactually surjective by the trivial identity Φ(Pcm) = Q and item a. above. Thus,Φ is a Banach space isomorphism. The L∞-linearity of Φ finally follows by conti-nuity and uniqueness of the extension of Φ and the density of SF(M ) in L∞(M , µ)w.r.t. the uniform norm after deriving the elementary identity

Φ 1B ω = 1B Φω

for every ω ∈ Pcm and every B ∈ B(M ) from (2.7).

2.2.2. Properties of the differential. Now we establish elementary calculus rulesassociated with the differential d from Definition 2.6.

We start with the following “closedness” property of it w.r.t. m [sic].

Lemma 2.10. For every sequence (fn)n∈N in Fe which converges to f ∈ L0(M )pointwise m-a.e. in such a way that the sequence (dfn)n∈N converges to ω ∈L2(T ∗M ) in L2(T ∗M ), we have f ∈ Fe as well as

df = ω. (2.8)

In particular, if fn ∈ F for every n ∈ N, fn f in L2(M ) and dfn ω inL2(T ∗M ) as n → ∞ for some f ∈ L2(M ) and ω ∈ L2(T ∗M ), then f ∈ F, and theidentity (2.8) holds accordingly.

Proof. Since (fn)n∈N is E-bounded by assumption, by Proposition 1.8 we directlyobtain that f ∈ Fe. Therefore, f − fm ∈ Fe for every m ∈ N, and again byProposition 1.8 and using that (fn)n∈N is E-Cauchy by (2.4),

limsupm→∞

‖d(f − fm)‖L2(T∗M) ≤ limsupm→∞

liminfn→∞

‖d(fn − fm)‖L2(T∗M) = 0.

It follows that dfm → df in L2(T ∗M ) as m → ∞, whence df = ω.The second claim is now due to Mazur’s lemma, up to possibly passing to suitable

pointwise m-a.e. converging subsequences.

“Expected” calculus rules for d hold if the measure µ under consideration isminimal E-dominant (recall from Subsection 1.3.5 that such µ does not charge E-polar sets). In particular, all identities in Proposition 2.11 below make sense by thedefinition of d in terms of µ and Proposition 1.12.

Proposition 2.11. Let µ be a minimal E-dominant Borel measure on M. Thenw.r.t. the L∞-module structure of L2(T ∗M ) from Theorem 2.4 induced by µ, thefollowing properties hold.

(i) Locality. For every f, g ∈ Fe,

1f=g

df = 1f=g

dg.

(ii) Chain rule. For every f ∈ Fe and every L1-negligible Borel set C ⊂ R,

1f−1(C)

df = 0.

In particular, for every ϕ ∈ Lip(R),

d(ϕ f) =[ϕ′ f

]df,

where the derivative ϕ′ f is defined arbitrarily on the intersection of theset of non-differentiability points of ϕ with the image of f .


(iii) Leibniz rule. For every f, g ∈ Feb(M ),

d(f g) = f dg + g df.

Proof. By the linearity of d, it is sufficient to consider the case where g vanishesidentically in point (i). Up to passing to a Borelian E-quasi-closed representative oftheE-quasi-closed set f = 0 [33, Lem. 2.5], by Theorem 2.4 we have 1

f=0df = 0

if and only if Γµ(f) = 0 µ-a.e. on f = 0. By Proposition 1.12, the assertion thusfollows from the identities

∥∥1f=0

df∥∥

L2(T∗M)= µf

[f = 0

]1/2= 0.

The first point of (ii) follows similar lines, taking Remark 1.15 into account.To prove the second claim in (ii), using Proposition 1.12 we note that

⟨d(ϕ f), dg

⟩µ

=⟨[ϕ′ f

]df, dg

⟩µ

µ-a.e.

for every g ∈ Fe. Lemma 2.7 allows us to extend this property to arbitrary ω ∈L2(T ∗M ) in place of dg. The Hilbert space structure of L2(T ∗M ) from Theorem 2.4,the fact that 〈·, ·〉µ induces the scalar product on L2(T ∗M ) by integration w.r.t. µas well as the arbitrariness of ω terminate the proof of (ii).

Item (iii) follows analogously to the previous argument.

Remark 2.12. If m is minimal E-dominant, all identities in Proposition 2.11 canequivalently be phrased with f and g in place of their E-q.c. m-versions f and g,respectively. In particular, we shall later refer to the corresponding modified versionof Proposition 2.11 without further comment.

2.3. Some remarks on the axiomatization.

Remark 2.13 (Compatibility with the smooth case). If M is a Riemannian manifoldwith boundary, in the setting of Example 1.18, L2(T ∗M ) coincides with the space ofL2-sections of the cotangent bundle T ∗M w.r.t. m, and d is the usual m-a.e. defineddifferential for, say, boundedly supported Lipschitz functions on M .

Remark 2.14 (Compatibility with [45]). By the m-a.e. equality between carré duchamp and minimal relaxed gradient outlined in Example 1.19, our approach isfully compatible with the one from [45, Sec. 2.2] for infinitesimally Hilbertian metricmeasure spaces (M , d,m).

Remark 2.15 (Compatibility with [10, 30, 36, 61, 62, 68]). The space H of 1-formsconstructed in [61, Ch. 2] agrees with L2(T ∗M ). For f ∈ F∩C0(M ), the differentialdf ∈ L2(T ∗M ) corresponds to the element f ⊗ 1M ∈ H in [61]. For instance, thisfollows by first proving that H is an L2-normed L∞-module w.r.t. an E-dominantµ [61, Ass. 2.1] which is generated by d(F∩ C0(M )), d := · ⊗ 1M , and applyingTheorem 2.9. The former is already done somewhat implicitly in [61, Ch. 2].

Remark 2.16 (Non-symmetric Dirichlet forms). The concepts presented above couldbe generalized to the case when E is non-symmetric in the sense of [78, Def. I.2.4]and satisfies the so-called strong sector condition from (I.2.4) in [78]. (The latteralways holds if E is symmetric.) See [72, 78] for further reading. In particular, by[80, Thm. 3.5] and owing to the transfer method [28], there exists a bilinear mapν : F2 → M

±fR(M )E such that

E(f, g) = νf,g[M ]

for every f, g ∈ F. Under the strong sector condition, ν can be extended to anon-relabeled bilinear map on F

2e by approximation. However,

ν = µ

32 MATHIAS BRAUN

holds on the diagonal of F2e , where solely in this remark, µ designates the en-

ergy measure of the symmetric part E of E, see (I.2.1) in [78]. In particular,the construction of L2(T ∗M ) — and hence also the one from Chapter 3 belowby Proposition 1.24 — only depends on E. In other words, even when starting witha non-symmetric form E, the cotangent module L2(T ∗M ) will be a Hilbert space.

This does not conflict with the not necessary infinitesimally Hilbertian settingfor metric measure spaces in [45, Ch. 2], although our approach is quite similarto [45], see Remark 2.14. The reason is that the underlying energy form in [45] isneither a priori induced by, nor a posteriori can be turned into a bilinear form in areasonable way. It is rather given in terms of the “diagonal energy measure” |D·|2 m,where |D · |2 is the so-called minimal relaxed gradient introduced in [4, Def. 4.2].Said differently, the Hilbertianity of L2(T ∗M ) in our setting simply comes since wehave started with a bilinear rather than merely a 2-homogeneous form.

3. Tangent module

3.1. Tangent vector fields and gradients. Now we dualize the concept of cotan-gent module introduced in Chapter 2 to define the so-called tangent module L2(TM ),i.e. an appropriate space of vector fields on M .

At the L∞-module level, different structures may arise depending on the choiceof E-dominant Borel measure µ in Theorem 2.4. In fact, although the tangentmodule L2(TM )µ with induced gradient ∇µ as in Definition 3.3 and Definition 3.4make perfect sense for every such µ, from now on we assume the following.

Assumption 3.1. E admits a carré du champ w.r.t. m.

Unless explicitly stated otherwise, all L∞-modules as well as their respectiveproperties in the sequel are understood w.r.t. m.

Remark 3.2. Besides Assumption 3.1 being worked under throughout Part 2, at thecurrent stage it has two further advantages. First, by Remark 2.12 the expectedcalculus rules from Proposition 2.11 hold and transfer accordingly to its induced gra-dient ∇. Second, Assumption 3.1 avoids incompatibilities between the divergenceoperator induced by different choices of (minimal) E-dominant reference measuresand the Laplacian ∆ associated to E, see Remark 3.6.

Definition 3.3. The tangent module (L2(TM ), ‖ · ‖L2(TM)) or simply L2(TM ) is

L2(TM ) := L2(T ∗M )∗

and it is endowed with the norm ‖ · ‖L2(TM) induced by (1.7). The elements ofL2(TM ) will be called vector fields.

As in Section 1.4, the pointwise pairing between ω ∈ L2(T ∗M ) and X ∈ L2(TM )is denoted by ω(X) ∈ L1(M ), and, by a slight abuse of notation, |X | ∈ L2(M )denotes the pointwise norm of such an X . By Lemma 2.7 and Proposition 1.24,L2(TM ) is a separable Hilbert module.

Furthermore, in terms of the pointwise scalar product 〈·, ·〉 on L2(T ∗M ) andL2(TM ), respectively, Proposition 1.24 allows us to introduce the musical isomor-phisms ♯ : L2(T ∗M ) → L2(TM ) and := ♯−1 defined by

⟨ω♯, X

⟩:= ω(X) =:

⟨X, ω

⟩m-a.e. (3.1)

Definition 3.4. The gradient of a function f ∈ Fe is defined by

∇f := (df)♯.


Observe from (3.1) that ∇f , where f ∈ Fe, is characterized as the unique elementX ∈ L2(TM ) which satisfies

df(X) = |df |2 = |X |2 m-a.e.

This uniqueness may fail on “nonlinear” spaces — that we do not consider here —such as Finsler manifolds. Compare with [45, Subsec. 2.3.1].

The gradient operator is clearly linear on Fe and closed in the sense of Lemma 2.10.By (3.1) and Proposition 2.11, all calculus rules from Proposition 2.11 transfer ac-cordingly to the gradient. Moreover, Lemma 2.7 ensures that L2(TM ) is generated,in the sense of L∞-modules, by ∇Fe and by ∇F.

3.2. Divergences. Now we introduce and study two notions of divergence of suit-able elements of L2(TM ). The first in Definition 3.5 is an L2-approach similarto [45, Subsec. 2.3.3]. The second in Definition 3.7 axiomatizes a measure-valueddivergence div. Both approaches are compatible in the sense of Lemma 3.11.

3.2.1. L2-divergence. The following is similar to [45, Def. 2.3.11]. See also [61,Ch. 3] for a similar construction for regular Dirichlet spaces.

Definition 3.5. We define the space D(div) to consist of all X ∈ L2(TM ) forwhich there exists a function f ∈ L2(M ) such that for every h ∈ F,

−ˆ

M

h f dm =ˆ

M

dh(X) dm.

In case of existence, f is unique, called the divergence of X and denoted by divX.

The uniqueness comes from the density of F in L2(M ). Note that div is a linearoperator on D(div), which thus turns D(div) into a vector space.

By (1.5), we have ∇D(∆) ⊂ D(div) and

div ∇f = ∆f m-a.e. (3.2)

for every f ∈ D(∆). Moreover, employing the Leibniz rule in Proposition 1.12, oneeasily can verify that for every X ∈ D(div) and every f ∈ Feb with |df | ∈ L∞(M ),we have f X ∈ D(div) and

div(f X) = f divX + df(X) m-a.e. (3.3)

Remark 3.6. In fact, (3.2) and (3.3) are the key reasons for Assumption 3.1 at thisfirst order level. Indeed, given a different E-dominant µ, the pairing 〈∇µf,X〉µ in(3.3) would require a possible divergence divµX for X ∈ L2(TM )µ to belong toL2(M , µ), while (3.2) somewhat forces the identity µ = m.

One way to bypass this if Assumption 3.1 does not hold is to regard the L2-divergence as simply acting on 1-forms instead of vector fields [61, Ch. 3].

3.2.2. Measure-valued divergence. The next definition has partly been inspired bythe work [23, Def. 4.1].

Definition 3.7. We define the space D(div) to consist of all X ∈ L2(TM ) forwhich there exists ν ∈ M

±σR(M )E such that for every h ∈ Fbc,

−ˆ

M

h dν =ˆ

M

dh(X) dm. (3.4)

In case of existence, ν is unique, termed the measure-valued divergence of X anddenoted by divX.

The uniqueness statement follows by density of Fbc in F by quasi-regularity ofE. The divergence div is clearly a linear operator on

DTV(div) :=X ∈ L2(TM ) : ‖divX‖TV < ∞

,

and the latter is a vector space.

34 MATHIAS BRAUN

3.2.3. Normal components and Gauß–Green’s formula. Before we proceed with var-ious properties of the notions from Definition 3.5 and Definition 3.7, it is convenientto introduce some further notation.

Definition 3.8. Given X ∈ D(div), its divergence div1 X ∈ L1loc(M ) and its nor-

mal component nX ∈ M±σR(M )E are defined by

div1 X :=ddiv≪ X

dm,

nX := − div⊥ X.

We define DL2(div) as the space of all X ∈ D(div) such that div1 X ∈ L2(M ).

Definition 3.8 is justified in the smooth setting according to the subsequentExample 3.9 and Remark 3.10.

Example 3.9. Let M be a Riemannian manifold with boundary ∂M . Recall thatwe denote by n the outward pointing unit normal vector field at ∂M . Then forevery smooth vector field X on M with compact support and every h ∈ C∞

c (M ),by Green’s formula, see e.g. [73, p. 44], we have

−ˆ

M

dh(X) dv =ˆ

M

h divvX dv −ˆ

∂M

h 〈X, n〉 ds.

Here, divvX is the usual divergence w.r.t. v. Thus, since C∞c (M ) is a core for F

(recall Example 1.18), X ∈ DTV(div) with

divX = divvX v-a.e.,

nX = 〈X, n〉 sX⊥

= 〈X⊥, n〉 s.

Hence nX is a measure which is supported on ∂M .If v is instead replaced by m := e−2w

v, w ∈ C2(M ), then for every X as above,in terms of the metric tensor 〈·, ·〉 we have X ∈ DTV(div) with

divX = divvX − 2 〈∇w,X〉 m-a.e.,

nX = e−2w 〈X, n〉 sX⊥

= e−2w 〈X⊥, n〉 s.

In fact, this smooth framework is the main motivation for the terminology ofDefinition 3.8, since in general we a priori do not know anything about the supportof nX , X ∈ D(div). As suggested by Example 3.9 and the following Remark 3.10,we may and will interpret nX as the normal component of X w.r.t. a fictive outwardpointing unit normal vector field — or more generally, as the “normal” part of Xw.r.t. a certain m-singular set. Examples which link our intrinsic approach tonormal components with existing extrinsic ones from [20, 23, 98] are discussed inAppendix A below.

As it turns out in Lemma 3.15 and Section 4.3, most vector fields of interest havevanishing normal component. Hence, normal components mostly do not appearin our subsequent treatise. However, our notion of second fundamental form inSubsection 8.2.4 really needs to make sense of a nonvanishing normal component ofcertain vector fields, see (8.12) and Example 8.24. This is precisely the motivationbehind Definition 3.7 and Definition 3.8, which have not been introduced in [45].In any case, in possible further applications it might be useful to see the normalcomponent of vector fields in D(div).


Remark 3.10 (Gauß–Green formula). In terms of Definition 3.7, given any X ∈D(div) and h ∈ Fbc, in analogy to Example 3.9 we have

−ˆ

M

dh(X) dm =ˆ

M

h divX dm −ˆ

M

h dnX.

3.2.4. Calculus rules. The proof of the following simple result directly follows fromthe respective definitions and is thus omitted.

Lemma 3.11. If X ∈ D(div) satisfies (divX)+ ∈ L1(M ) or (divX)− ∈ L1(M ),then X ∈ DL2 (div) and

div1 X = divX m-a.e.,

nX = 0.

Conversely, if X ∈ DL2(div) with nX = 0, then X ∈ D(div) with

divX = div1 X m-a.e.

Remark 3.12. The additional integrability condition in the first part of Lemma 3.11ensure that divX m is a well-defined signed Borel measure for X ∈ D(div). In fact,unlike (3.2) this small technical issue prevents us from saying that in general, iff ∈ D(∆) then ∇f ∈ DL2 (div) with div1 ∇f = ∆f m-a.e. and n ∇f = 0. Thiswould correspond to the role of ∆ as Neumann Laplacian from Subsection 1.3.6.

However, by the integration by parts Definition 3.5 as well as Lemma 3.13 andLemma 3.15 below there is still formal evidence in keeping this link in mind. Inparticular, we may and will interpret every X ∈ D(div) as having vanishing normalcomponent although the natural assignment divX := divX m — which entailsnX = 0 — might not be well-defined in M

±σ (M )E.

Based upon Lemma 3.11, as long as confusion is excluded we make no furthernotational distinction and identify, for suitable X ∈ L2(TM ),

divX = div1 X.

Lemma 3.13. For every X ∈ D(div) and every f ∈ Feb such that |X | ∈ L∞(M )or |df | ∈ L∞(M ), we have f X ∈ D(div) with

div(f X) = f divX + df(X)m,

fdiv(f X) = f divX + df(X) m-a.e.,

n(f X) = f nX.

In particular, if X ∈ DL2(div) then also f X ∈ DL2(div).

Proof. The first identity, from which the second follows, is straightforward to deducefrom Proposition 1.12. The last claim on DL2(div) is a direct consequence of thesetwo identities. To prove the remaining claim n(f X) = f nX, we compute

n(f X) = − div⊥(f X)

= f divX m − f divX

= −f div⊥ X

= f nX.

Example 3.14. Every identity stated in Lemma 3.13 is fully justified in the smoothcontext of Example 3.9. In particular, for every smooth vector field X on M withcompact support and every f ∈ C∞

c (M ),

n(f X) = 〈f X, n〉 s = f 〈X, n〉 s = f nX.

36 MATHIAS BRAUN

Finally, we show that Lemma 3.11 is not void, i.e. that there will exist an L2-dense set of vector fields — see Section 4.3 below — which satisfy both hypothesesof Lemma 3.11 even in a slightly stronger version.

Lemma 3.15. Suppose that f ∈ D(∆) and g ∈ Fb. Moreover, suppose that |∇f | ∈L∞(M ) or |dg| ∈ L∞(M ). Then g∇f ∈ DTV(div) ∩ D(div) with

div(g∇f) = dg(∇f) + g∆f m-a.e.,

n(g∇f) = 0.

Proof. By (3.3) and (3.2), we already know that g∇f ∈ D(div).To show that g∇f ∈ DTV(div), note that under the given assumptions, the

r.h.s. of the identity for div(g∇f) belongs to L1(M ). In particular, by what wehave already proved before, ν := divX m ∈ M

±f (M )E satisfies the defining property

(3.4) for divX , yielding the claim.

We conclude this survey with a duality formula. Recall Remark 3.12 to see whyit might be hard to verify in general that the second inequality is an equality.

Proposition 3.16. For every f ∈ F,

E(f) = sup

−2ˆ

M

f divX dm −ˆ

M

|X |2 dm : X ∈ D(div)

≥ sup

−2ˆ

M

f divX dm −ˆ

M

|X |2 dm : X ∈ DL2(div), nX = 0.

Moreover, if M has finite m-measure, the second inequality is an equality.

Proof. The second inequality is a trivial consequence of Lemma 3.11.Moreover, if m[M ] < ∞, by Lemma 3.11 every X ∈ D(div) belongs to DL2 (div)

with divX = divX m, whence nX = 0. This yields the last statement.Let us finally turn to the first equality.To prove “≥”, given any X ∈ D(div), it follows from Young’s inequality and

Definition 3.5 that

E(f) +ˆ

M

|X |2 dm ≥ 2ˆ

M

df(X) dm = −2ˆ

M

f divX dm.

Rearranging terms and taking the supremum over all X as above directly impliesthe desired inequality.

To prove the converse inequality, given any f ∈ F and any t > 0, we set Xt :=∇ptf ∈ L2(TM ). Since ptf ∈ D(∆), it follows from (3.2) that Xt ∈ D(div) withdivXt = ∆ptf m-a.e. Using the semigroup property of (pt)t≥0, (1.5) and thenonincreasingness of the function t 7→ E(ptf) on [0,∞), we arrive at

limsupt→0

[− 2ˆ

M

f divXt dm −ˆ

M

|Xt|2 dm]

= limsupt→0

[E(ptf) −

ˆ

M

f∆ptf dm −ˆ

M

|∇ptf |2 dm]

= limsupt→0

[E(ptf) +

ˆ

M

|∇pt/2f |2 dm −ˆ

M

|∇ptf |2 dm]

≥ limsupt→0

E(ptf) = E(f).ˆ

M

The desired inequality readily follows.


Part 2. Second order differential structure

Throughout this part, let Assumption 3.1 hold. We employ the following con-vention: given f ∈ L0(M ), integrals of possibly degenerate terms such as 1/f are(consistently) understood as restricted on f 6= 0 without further notice.

4. Preliminaries. The taming condition

4.1. Basic notions of tamed spaces. In this section, we outline the theory oftamed spaces introduced in [41] that will be needed below throughout.

4.1.1. Quasi-local distributions. Given an E-quasi-open set G ⊂ M , let F−1G denote

the dual space of the closed [29, p. 84] subspace

FG :=f ∈ F : f = 0 E-q.e. on Gc

of F. Note that if G′ ⊂ M is E-quasi-open as well with G ⊂ G′, then F−1G ⊃ F

−1G′ .

Let F−1qloc denote the space of E-quasi-local distributions on F, i.e. the space of

all objects κ for which there exists an E-quasi-open E-nest (Gn)n∈N such thatκ ∈ ⋂

n∈NF

−1Gn

. Every distribution κ ∈ F−1qloc is uniquely associated with an additive

functional, or briefly AF, (aκt )t≥0 in the sense of [41, Lem. 2.7, Lem. 2.9]. Here,uniqueness means up to m-equivalence of AF’s [43, p. 423]. (We refer to [43, Ch. II]or [29, Ch. 4, App. A] for a concise overview over basic notions about AF’s.) TheAF (aκt )t≥0 is independent of the chosen E-nest [41, Lem. 2.11]. All AF’s will beunderstood as being zero beyond the explosion time ζ.

Remark 4.1. (The defining properties of) AF’s are linked to the Markov process M

from Subsection 1.3.8 [29, Def. A.3.1].

Example 4.2. If κ ∈ F−1qloc is induced in the evident way through a nearly Borel [29,

Def. A.1.28] function f ∈ L2(M ), by [41, Rem. 2.8], for every t ∈ [0, ζ),

aκt =12

ˆ 2t

0

f(bs) ds.

4.1.2. Extended Kato class. The relevant distributions κ ∈ F−1qloc in our work, see

Remark 4.6, will be induced by signed measures in the extended Kato class K1−(M ),which is introduced now. In fact, Feynman–Kac semigroups and energy formsinduced by more general distributions κ ∈ F

−1qloc are studied in [41].

Given any Borelian f : M → R, set

q-sup f := inf

supx∈(M ′)c

f(x) : M ′ ⊂ M is E-polar.

Recall from [29, p. 84] that a measure ν ∈ M+(M ) is E-smooth if it does not charge

E-polar sets and ν[Fn] < ∞ for every n ∈ N, for some E-nest (Fn)n∈N of closedsets. Every Radon measure charging no E-polar set is E-smooth, but the conversedoes not hold [33, Ex. 2.33]. By the Revuz correspondence [29, Thm. A.3.5], anyE-smooth ν ∈ M

+(M ) is uniquely associated to a positive continuous AF, or brieflyPCAF, (aνt )t≥0 by the subsequent identity, valid for every nonnegative f ∈ L0(M ):

ˆ

M

f dν = limt→0

1t

ˆ

M

E ·[ˆ t

0

f(b2s) daνs

]dm.

Of course, the existence of a compact E-nest, by quasi-regularity of E, ensuresthat every σ-finite ν ∈ M

+(M ) charging no E-polar set is E-smooth. Hence thefollowing definition [41, Def. 2.21] is meaningful.

38 MATHIAS BRAUN

Definition 4.3. Given ρ ≥ 0, the ρ-Kato class Kρ(M ) of M is defined to consistof all µ ∈ M

±σ (M ) which do not charge E-polar sets with

limt→0

q-sup E ·[a

2|µ|t

]≤ ρ.

K0(M ) is called Kato class of M, while the extended Kato class of M is

K1−(M ) :=⋃

ρ∈[0,1)

Kρ(M ).

Remark 4.4. By definition, we have µ ∈ Kρ(M ) if and only if |µ| ∈ Kρ(M ) if andonly if µ+, µ− ∈ Kρ(M ), ρ ≥ 0.

The following important lemma is due to [41, Cor. 2.25]. The immediate corollarythat one can always choose ρ′ < 1 for µ ∈ K1−(M ) therein is used crucially in ourwork (and also in [41]), see e.g. Corollary 5.12 and Lemma 8.8.

Lemma 4.5. For every ρ, ρ′ ≥ 0 with ρ < ρ′ and every µ ∈ Kρ(M ) there existsα′ ∈ R such that for every f ∈ F,

ˆ

M

f2 dµ ≤ ρ′E(f) + α′

ˆ

M

f2 dm.

In particular, w.r.t. a sequence (Gn)n∈N of open subsets of M witnessing theσ-finiteness of ν := |µ|, by Cauchy–Schwarz’s inequality every µ ∈ Kρ(M ), ρ ≥ 0,induces a (non-relabeled) element µ ∈ F

−1qloc(M ) by setting, for f ∈ ⋃

n∈NFGn

,⟨µ

∣∣ f⟩

:=ˆ

M

f dµ.

If µ ∈ Kρ(M ) is nonnegative, then the AF’s coming from the Revuz correspondencew.r.t. µ and from its property as inducing an element in F−1

qloc are m-equivalent.The key feature about general µ ∈ K1−(M ) is that by Khasminskii’s lemma [41,Lem. 2.24], the induced distribution is moderate [41, Def. 2.13], i.e.

supt∈[0,1]

q-sup E ·[e−a

2µt

]< ∞.

4.1.3. Feynman–Kac semigroup. Let q ∈ [1, 2], and note that qκ/2 ∈ K1−(M ) forevery κ ∈ K1−(M ). Given such κ, we define a family (pqκt )t≥0 of operators actingon nonnegative nearly Borel functions f ∈ L0(M ) by

pqκt f := E ·

[e−a

qκt f(b2t) 1t<ζ/2

].

It naturally extends to nearly Borelian f ∈ L0(M ) for which the latter expectationfor |f | in place of f is finite, see [41, Def. 2.10]. (For later convenience, we have tochange the notation from [41] a bit, also at later times, see Remark 4.7 below.) It ism-symmetric and maps m-equivalence classes to m-equivalence classes. Since qκ/2is moderate, it extends to an exponentially bounded semigroup of linear operatorson Lp(M ) for every p ∈ [1,∞] [41, Lem. 2.11, Rem. 2.14]. That is, there exists afinite constant C > 0 such that for every p ∈ [1,∞] and every t ≥ 0,

∥∥pqκt

∥∥Lp(M);Lp(M)

≤ C eCt.

4.1.4. The perturbed energy form. One of the main results from [41] is that forκ ∈ K1−(M ) — in fact, for more general κ ∈ F

−1qloc, cf. [41, Thm. 2.49] — (pqκt )t≥0

is properly associated to an energy form Eqκ, q ∈ [1, 2]. Indeed, by [41, Thm. 2.47,

Thm. 2.49, Cor. 2.51], the quadratic form

Eqκ(f) := E(f) + q

⟨κ

∣∣ f2⟩

(4.1)

with finiteness domain D(Eqκ) = F is closed, lower semibounded and associatedwith (pqκt )t≥0 [43, Thm. 1.3.1, Lem. 1.3.2].


The corresponding generator, henceforth termed ∆qκ with domain D(∆qκ), iscalled Schrödinger operator with potential qκ.

Remark 4.6. One reason for considering perturbations of E by κ ∈ K1−(M ) is thefollowing. By Lemma 4.5, the map f 7→

⟨κ−

∣∣ f2⟩

on F is form bounded w.r.t. E,hence w.r.t. Eqκ

+

, with some form bound ρ′ < 1. Hence, by [41, Thm. 2.49], Eqκ isclosed with domain D(Eqκ) =

f ∈ F :

⟨κ+

∣∣ f2⟩< ∞

. Again by Lemma 4.5, the

latter is all of F, which is technically required in the setting of Section 4.2, comparewith [41, Ch. 6]. See also Remark 4.11 below.

Remark 4.7. For our analytic and geometric purposes, we use differently scaledforms, operators and semigroups than [41]. Let us list the relations of the mainobjects in [41], on the l.h.s.’s, with our notation, on the respective r.h.s.’s:

E = E/2,∣∣

L = ∆/2,∣∣

Pt = pt/2,∣∣

Bt = bt [sic],∣∣

Eqκ/2 = E

qκ/2∣∣

Lqκ/2 = ∆qκ/2,

Pqκ/2t = p

qκt/2.

4.1.5. Tamed spaces.

Definition 4.8. Suppose that q ∈ 1, 2, κ ∈ K1−(M ) and N ∈ [1,∞]. We say that(M ,E,m) or simply M satisfies the q-Bakry–Émery condition, briefly BEq(κ,N),if for every f ∈ D(∆) with ∆f ∈ F and every nonnegative φ ∈ D(∆qκ) with∆qκφ ∈ L∞(M ), we have

1q

ˆ

M

∆qκφ |∇f |q dm −ˆ

M

φ |∇f |q−1⟨∇f,∇∆f

⟩dm

≥ 1N

ˆ

M

φ |∇f |q−1 (∆f)2 dm.

The latter term is understood as 0 if N := ∞.

Remark 4.9. Through a mollification argument using (pqκt )t≥0 [41, Lem. 6.2], theclass of elements φ ∈ D(∆qκ) with ∆qκφ ∈ L∞(M ) is dense in L2(M ).

Assumption 4.10. We henceforth assume that M satisfies the BE2(κ,N) condi-tion for given κ ∈ K1−(M ) and N ∈ [1,∞].

For certain k : M → R and N ∈ [1,∞], write BE2(k,N) instead of BE2(km, N).Of course, Definition 4.8 [41, Def. 3.1, Def. 3.5] generalizes the well-known Bakry–Émery condition for uniform lower Ricci bounds [6, 8, 9, 40, 44]. Variable lowerRicci bounds have been first studied by [17, 95] in a synthetic context.

In the framework of Assumption 4.10, we say that (M ,E,m) or simply M istamed. Although this is not the original definition of taming from [41, Def. 3.2], forκ ∈ K1−(M ) they are in fact equivalent, see Section 4.2 below.

Remark 4.11. From the taming point of view, one might regard the implicit as-sumption that κ+ ∈ K1−(M ) (recall Remark 4.4) as unnatural. However, for thequalitative message of this article — the existence of a rich second order calculus— one can simply ignore κ+ by setting it to zero. To obtain quantitative resultsin applications, to bypass the assumption κ+ ∈ K1−(M ), a useful tool could beappropriate “cutoffs” and monotone approximations by elements in K1−(M ), seee.g. Subsection 8.4.2 below and [17, Lem. 2.1].

40 MATHIAS BRAUN

Example 4.12 (Manifolds). Any compact Riemannian manifold M is tamed by

κ := k v + ls

which in fact belongs to K0(M ) [41, Thm. 4.4] (recall Example 1.18). More gen-erally [15, 18], let M be a “regular” Lipschitz Riemannian manifold, in the senseof [18], that is quasi-isometric to a Riemannian manifold with uniformly lowerbounded Ricci curvature. Suppose that the Ricci curvature of M , where defined, isbounded from below by a function k ∈ Lp(M ,Ξ v), p > d/2, where Ξ: M → R isgiven by Ξ(x) := v[B1(x)]−1. Then M is tamed by κ := k v ∈ K0(M ).

Example 4.13 (RCD spaces). Every RCD(K,∞) space (M , d,m), K ∈ R, accordingto Example 1.19 is tamed by κ := K m ∈ K0(M ) [5] (recall Example 4.2).

Example 4.14 (Almost smooth spaces). Let (M , d,m) be a d-dimensional almostsmooth metric measure space, d ∈ N, in the sense of [65, Def. 3.1, Def. 3.16],examples of which include the gluing of two pointed, compact Riemannian manifolds(not necessarily of the same dimension) at their base points. Then, under fewfurther assumptions, Honda proved that if the “generalized Ricci curvature” of Mis bounded from below by K(d− 1), K ∈ R, then the BE2(K(d− 1), d) conditionholds for M [65, Thm. 3.7, Thm. 3.17]. The RCD∗(K(d−1), N) condition, however,does not hold in general [65, Rem. 3.9].

Example 4.15 (Configuration spaces). Further important nonsmooth examples areconfiguration spaces Y over Riemannian manifolds M [1, 39]. The Dirichlet formE

Y on Y constructed in [1] is quasi-regular and strongly local, cf. the proof of [1,Thm. 6.1]. If Ric ≥ K on M , K ∈ R, then (Y,EY , π) is tamed by κ := K π ∈ K0(M )as well by [40, Thm. 4.7] and [41, Thm. 3.6]. Here π is the Poisson (probability)measure on Y, up to intensity.

A similar result even over more general spaces is announced in [33].

Spaces that are tamed by some measures in K1−(M ) or even K0(M ) may alsohave cusp-like singularities [41, Thm. 4.6], have singular [41, Thm. 2.36] or notsemiconvex boundary [41, Thm. 4.7] or be of Harnack-type [18, 41].

4.1.6. Intrinsically complete Dirichlet spaces. An interesting, but not exhaustive,class of tamed spaces we sometimes consider is the one of intrinsically complete Mas introduced in [41, Def. 3.8].

Definition 4.16. We call (M ,E,m) or simply M intrinsically complete if thereexists a sequence (φn)n∈N in F such that m

[φn > 0

]< ∞, 0 ≤ φn ≤ 1 m-a.e. and

|∇φn| ≤ 1 m-a.e. for every n ∈ N as well as φn → 1M and |∇φn| → 0 pointwisem-a.e. as n → ∞.

Intrinsically complete tamed spaces are stochastically complete, i.e.

pt1M = 1M m-a.e.

for every t ≥ 0 [41, Thm. 3.11]. In our work, intrinsically complete spaces providea somewhat better version of Lemma 1.9, sometimes used to get rid of differentialsin certain expressions. See e.g. Remark 6.7 and Remark 7.3. However, none of ourresults will severely rely on intrinsic completeness.

4.2. Self-improvement and singular Γ2-calculus. In fact, under the aboveAssumption 4.10, the condition BE2(κ,N) is equivalent to BE1(κ,N) [41, Thm. 6.9],N ∈ [1,∞], albeit the latter is a priori stronger [41, Thm. 3.4, Prop. 3.7]. In par-ticular, the heat flow (pt)t≥0 satisfies the important contraction estimate

|∇ptf | ≤ pκt |∇f | m-a.e. (4.2)


for every f ∈ Fand every t ≥ 0. See [8, 9, 87] for corresponding results for constantκ and [17, Thm. 3.6] for the first nonconstant result in this direction.

We briefly recapitulate the singular Γ2-calculus developed in [41, 87], since the in-volved calculus objects, in particular those from Subsection 4.2.1 below, are crucialin our treatise as well, see e.g. Theorem 5.11 and Theorem 8.9.

4.2.1. Test functions. Define the set of test functions by

Test(M ) :=f ∈ D(∆) ∩ L∞(M ) : |∇f | ∈ L∞(M ), ∆f ∈ F

It is an algebra w.r.t. pointwise multiplication, and if f ∈ Test(M )n and ϕ ∈C∞(Rn) with ϕ(0) = 0, n ∈ N, then ϕ f ∈ Test(M ) as well [87, Lem. 3.2].

Since BE2(κ,N) implies BE2(−κ−, N) [41, Prop. 6.7], a variant of the reversePoincaré inequality states that for every f ∈ L2(M ) ∩ L∞(M ) and every t > 0,

|∇ptf |2 ≤ 12t

∥∥p−2κ−

t

∥∥L∞(M);L∞(M)

‖f‖L∞(M) m-a.e.,

and hence ptf ∈ Test(M ) [41, Cor. 6.8]. In particular, Test(M ) is dense in F.We use the following two approximation results henceforth exploited at various

instances. For convenience, we outline the proof of Lemma 4.17. Lemma 4.18,which yields a useful density result for the slightly smaller set

TestL∞(M ) :=f ∈ Test(M ) : ∆f ∈ L∞(M )

,

results from (4.2) and an approximation by a mollified heat flow [41, 87].

Lemma 4.17. For every f ∈ Test(M ), there exist sequences (gn)n∈N and (hn)n∈N

in Test(M ) which are bounded in L∞(M ) with gn hn → f in F as n → ∞.

Proof. Given any k,m ∈ N, we define gk := 2 arctan(k f)/π ∈ Test(M ) and hm :=(f2 + 2−m)1/2 − 2−m/2 ∈ Test(M ). Then gk hm → gk |f | in L2(M ) as m → ∞ forevery k ∈ N, and gk |f | → f in L2(M ) as k → ∞. Using Proposition 2.11 andLebesgue’s theorem, it follows that gk hm → gk |f | in F as m → ∞ for every k ∈ N,and gk |f | → f in F as k → ∞. We conclude by a diagonal argument.

Lemma 4.18. For every f ∈ F such that a ≤ f ≤ b m-a.e., a, b ∈ [−∞,∞],there exists a sequence (fn)n∈N in TestL∞(M ) which converges to f in F such thata ≤ fn ≤ b m-a.e. for every n ∈ N. If moreover |∇f | ∈ L∞(M ), then (fn)n∈N canbe constructed such that (|∇fn|)n∈N is bounded in L∞(M ).

4.2.2. Measure-valued Schrödinger operator. A further regularity property of func-tions f ∈ Test(M ) that will be crucial in defining the κ-Ricci measure in Section 8.1is that their carré du champs |∇f |2 have F-regularity under BE2(κ,N). In fact,|∇f |2 admits a measure-valued Schrödinger operator in the sense of Definition 4.19.This is recorded in Proposition 4.20 and is due to [41, Lem. 6.4].

Definition 4.19. We define D(∆2κ) to consist of all u ∈ F for which there existsι ∈ M

±σR(M )E such that for every h ∈ F, we have h ∈ L1(M , ι) and

ˆ

M

h dι = −E2κ(h, u).

In case of existence, ι is unique, denoted by ∆2κu and shall be called the measure-valued Schrödinger operator with potential 2κ.

Proposition 4.20. For every f ∈ Test(M ) we have |∇f |2 ∈ F and even |∇f |2 ∈D(∆2κ). Moreover,

12

∆2κ|∇f |2 −⟨∇f,∇∆f

⟩m ≥ 1

N(∆f)2

m.

42 MATHIAS BRAUN

An advantage of interpreting the Schrödinger operator associated to E2κ as a

measure is that the potential 2κ can be separated from ∆2κ to give the measure-valued Laplacian ∆ := ∆2κ + 2κ, which fits well with the divergence objects fromSection 3.2, see Subsection 8.2.1 below. This is technically convenient in definingthe drift-free Ricci measure Ric in Definition 8.16 and the second fundamental formII in Definition 8.23 without κ-dependency. However, for possible later extensions,e.g. when κ is not a (signed) measure, we decided not to separate the distributionκ from the other calculus objects under consideration until Subsection 8.2.1.

4.2.3. Singular Γ2-operator. Given Proposition 4.20, following [41, 87] we introducethe map Γ2κ

2 : Test(M ) → M+σR(M )E by

Γ2κ2 (f) :=

12

∆2κ|∇f |2 −⟨∇f,∇∆f

⟩m. (4.3)

According to the Lebesgue decomposition in Subsection 1.1.1, we decompose

Γ2κ2 (f) = Γ2κ

2 (f)≪ + Γ2κ2 (f)⊥

w.r.t. m, f ∈ Test(M ). A consequence of Proposition 4.20 is that

Γ2κ2 (f)⊥ ≥ 0

and, defining γ2κ2 (f) := dΓ2κ

2 (f)≪/dm ∈ L1(M ),

γ2κ2 (f) ≥ 1

N(∆f)2

m-a.e.

Further calculus rules of Γ2κ2 are summarized in the next Lemma 4.21. To this aim,

note that 〈∇u,∇v〉 ∈ F for every u, v ∈ Test(M ) by Proposition 4.20, whence itmakes sense define the “pre-Hessian” H[·] : Test(M )3 → L2(M ) by

2 H[f ](g1, g2) :=⟨∇g1,∇〈∇f,∇g2〉

⟩+

⟨∇g2,∇〈∇f,∇g1〉

⟩

−⟨∇f,∇〈∇g1,∇g2〉

⟩.

(4.4)

Lemma 4.21. Let α ∈ N, q ∈ Test(M )α and ϕ ∈ C∞(Rα) with ϕ(0) = 0.Moreover, given any i, j ∈ 1, . . . , α, set ϕi := ∂iϕ and ϕij := ∂i∂jϕ. DefineA2κ[ϕ q] ∈ M

±f (M )E and B[ϕ q],C[ϕ q],D[ϕ q] ∈ L1(M ) by

A2κ[ϕ q] :=α∑

i,j=1

[ϕi q

] [ϕj q

]Γ2κ

2 (qi, qj),

B[ϕ q] := 2α∑

i,j,k=1

[ϕi q

] [ϕjk q] H[qi](qj , qk),

C[ϕ q] :=α∑

i,j,k,l=1

[ϕik q

] [ϕjl q

]〈∇qi,∇qj〉〈∇qk,∇ql〉,

D[ϕ q] :=[ α∑

i=1

[ϕi q

]∆qi +

α∑

i,j=1

[ϕij q] 〈∇qi,∇qj〉

]2

.

Then we have the identities

Γ2κ2 (ϕ q) = A2κ[ϕ q] +

[B[ϕ q] + C[ϕ q]

]m,

[∆(ϕ q)

]2m = D[ϕ q]m.

Remark 4.22. Note that all measures in Lemma 4.21 are identified as finite. Thistechnical point is shown in Proposition 4.24 below. Albeit the latter is an a poste-riori consequence of BE1(κ,∞), see also Remark 4.25, the results of [41, Lem. 6.5,Thm. 6.6, Thm. 6.9] — in particular Lemma 4.21 — are still deducible a priorifrom Assumption 4.10 by restriction of the identities from [41, Lem. 6.5, Thm. 6.6]


to subsets of finite measure, or interpreting the asserted identities in a suitable weaksense.

4.2.4. Finiteness of total variations. The final goal of this subsection is to provein Proposition 4.24 that ‖∆2κ|∇f |2‖TV < ∞, f ∈ Test(M ). Besides the technicalRemark 4.22, this fact will be of decisive help in continuously extending the κ-Riccimeasure beyond regular vector fields, see Theorem 8.9.

The following is a minor variant of [41, Lem. 6.2] with potential κ instead of 2κ,proven in a completely analogous way.

Lemma 4.23. Let u ∈ L2(M ) ∩ L∞(M ) be nonnegative, and let g ∈ L2(M ). Sup-pose that for every nonnegative φ ∈ D(∆κ) ∩ L∞(M ) with ∆κφ ∈ L∞(M ),

ˆ

M

u∆κφdm ≥ −ˆ

M

g φdm.

Then u ∈ F as well as

Eκ(u) ≤

ˆ

M

u g dm.

Moreover, there exists a unique measure σ ∈ M+σ (M )E such that for every h ∈ F,

we have h ∈ L1(M ,σ) andˆ

M

h dσ = −Eκ(h, u) +

ˆ

M

h g dm.

Proposition 4.24. For every f ∈ Test(M ), |∇f | belongs to Fb, and the signedBorel measure ∆2κ|∇f |2 has finite total variation. Moreover,

Γ2κ2 (f)[M ] =

ˆ

M

(∆f)2 dm −⟨κ

∣∣ |∇f |2⟩,

∆2κ|∇f |2[M ] = −2⟨κ

∣∣ |∇f |2⟩.

(4.5)

Proof. By Proposition 4.20, we already know that |∇f |2 ∈ D(∆2κ). Now recallthat by the self-improvement property of BE2(κ,N), (M ,E,m) obeys BE1(κ,∞)according to Definition 4.8 [41, Thm. 6.9]. By Lemma 4.23 applied to u := |∇f | ∈L2(M ) ∩ L∞(M ) and g := −1|∇f |>0 〈∇f,∇∆f〉 |∇f |−1 ∈ L2(M ) as well as by(4.1), we obtain |∇f | ∈ Fb and the unique existence of an element σ ∈ M

+σR(M )E

such that for every h ∈ F, we have h ∈ L1(M ,σ) andˆ

M

h dσ = −Eκ(h, |∇f |

)−ˆ

|∇f |>0

h⟨∇f,∇∆f

⟩|∇f |−1 dm. (4.6)

Inserting h := φ |∇f | for φ ∈ Fb in (4.6) yieldsˆ

M

φ |∇f |∼ dσ = −Eκ(φ |∇f |, |∇f |

)−ˆ

M

φ⟨∇f,∇∆f

⟩dm.

Hence for such φ and using the Definition 4.19 of ∆2κ,ˆ

M

φ d∆2κ|∇f |2 = −E2κ

(φ, |∇f |2

)

= −2ˆ

M

|∇f |⟨∇φ,∇|∇f |

⟩dm − 2

⟨κ

∣∣φ |∇f |2⟩

= −2Eκ(φ |∇f |, |∇f |

)+ 2ˆ

M

φ∣∣∇|∇f |

∣∣2dm

= 2ˆ

M

φ |∇f |∼ dσ + 2ˆ

M

φ⟨∇f,∇∆f

⟩dm

+ 2ˆ

M

φ∣∣∇|∇f |

∣∣2dm.

44 MATHIAS BRAUN

Since φ ∈ Fb is arbitrary, we get

∆2κ|∇f |2 = 2 |∇f |∼ σ + 2⟨∇f,∇∆f

⟩m + 2

∣∣∇|∇f |∣∣2m. (4.7)

Indeed, the r.h.s. is well-defined since 〈∇f,∇∆f〉m and∣∣∇|∇f |

∣∣2m define (signed)

Borel measures of finite total variation. Setting h := |∇f | in (4.6) implies that|∇f |∼ σ is finite as well, whence ∆2κ|∇f |2 is of finite total variation.

Finally, the second identity from (4.5) follows from combining (4.7) with (4.6)for h := |∇f |, which in turn gives the first identity by the definition (4.3).

Remark 4.25. On RCD(K,∞) spaces (M , d,m), K ∈ R, according to Example 1.19the argument for the finiteness of ‖∆2K |∇f |2‖TV is more straightforward: it followsby conservativeness of the heat flow (pt)t≥0 [5, 6] and does not require the detourover the BE1(K,∞) condition [87, Lem. 2.6]. In our work, conservativeness isneither assumed nor generally a consequence of Definition 4.8 (recall Remark 1.11).

Remark 4.26 (Caveat). The relation (4.6) suggests to derive that |∇f | ∈ D(∆κ),with ∆κ defined appropriately as in Definition 4.19, with

∆κ|∇f | = σ +⟨∇f,∇∆f

⟩|∇f |−1

m.

However, it is not clear if the r.h.s. defines an element of M±σR(M ) since neither the

summands on the r.h.s. typically define finite measures, nor we really know whetherthe last summand is a signed measure (it might take both the value ∞ and −∞).

The situation changes when treating the L1-Bochner inequality for test vectorfields, see Theorem 8.28 below.

4.3. Lebesgue spaces and test objects. This section is a survey over the defi-nition of the spaces Lp(T ∗M ) and Lp(TM ), p ∈ [1,∞], of p-integrable (co-)tangentvector fields w.r.t. m.

4.3.1. The L0-modules L0(T ∗M ) and L0(TM ). Let L0(T ∗M ) and L0(TM ) be theL0-modules as in Subsection 1.4.3 associated to L2(T ∗M ) and L2(TM ), i.e.

L0(T ∗M ) := L2(T ∗M )0,

L0(TM ) := L2(TM )0.

The characterization of Cauchy sequences in these spaces [45, p. 31] grants thatthe pointwise norms | · | : L2(T ∗M ) → L2(M ) and | · | : L2(TM ) → L2(M ) as wellas the musical isomorphisms : L2(TM ) → L2(T ∗M ) and ♯ : L2(T ∗M ) → L2(TM )uniquely extend to (non-relabeled) continuous maps | · | : L0(T ∗M ) → L0(M ), | ·| : L0(TM ) → L0(M ), : L0(TM ) → L0(T ∗M ) and ♯ : L0(T ∗M ) → L0(M ). (Andthe latter two will restrict to pointwise isometric module isomorphisms between therespective Lp-spaces, p ∈ [1,∞], from Subsection 4.3.2.)

4.3.2. The Lebesgue spaces Lp(T ∗M ) and Lp(TM ). For p ∈ [1,∞], let Lp(T ∗M )and Lp(TM ) be the Banach spaces consisting of all ω ∈ L0(T ∗M ) and X ∈ L0(TM )such that |ω| ∈ Lp(M ) and |X | ∈ Lp(M ), respectively, endowed with the norms

‖ω‖Lp(T∗M) :=∥∥|ω|

∥∥Lp(M)

,

‖X‖Lp(TM) :=∥∥|X |

∥∥Lp(M)

.

Since by Lemma 2.7, L2(T ∗M ) is separable — and so is L2(TM ) by Proposition 1.24— one easily derives that if p < ∞, the spaces Lp(T ∗M ) and Lp(TM ) are separableas well. Since L2(T ∗M ) and L2(TM ) are reflexive as Hilbert spaces, by the discus-sion from Subsection 1.4.2 it follows that Lp(T ∗M ) and Lp(TM ) are reflexive for


every p ∈ [1,∞], and that for q ∈ [1,∞] such that 1/p+ 1/q = 1, in the sense ofL∞-modules we have the duality

Lp(T ∗M )∗ = Lq(TM ).

4.3.3. Test and regular objects. As in [45, p. 102], using Lemma 4.18 we see thatthe linear span of all elements of the form h∇g, g ∈ TestL∞(M ) and h ∈ Test(M ),is weakly∗ dense in L∞(TM ). This motivates to consider the subsequent subclassesof L2(TM ) consisting of test vector fields or regular vector fields, respectively:

TestL∞(TM ) := n∑

i=1

gi ∇fi : n ∈ N, fi, gi ∈ TestL∞(M ),

Test(TM ) := n∑

i=1

gi ∇fi : n ∈ N, fi, gi ∈ Test(M ),

Reg(TM ) := n∑

i=1

gi ∇fi : n ∈ N, fi ∈ Test(M ), gi ∈ Test(M ) ∪ R 1M

.

Remark 4.27. These three classes play different roles in the sequel. TestL∞(TM ) isjust needed for technical reasons when some second order L∞-control is required.Test(TM ) is usually the class of vector fields w.r.t. which certain objects are definedby testing against, while Reg(TM ) is the typical class of vector fields for which suchobjects are defined. We make this distinction between Test(TM ) and Reg(TM ),which has not been done in [45], for the reason that we want to include bothvector fields with “regular” zeroth order part as well as pure gradient vector fieldsfor differential objects such as the covariant derivative, see Theorem 6.3, or theexterior differential, see Theorem 7.5. However, under the usual closures that wetake below, it is not clear if gradient vector fields belong to those w.r.t. test ratherthan regular objects. Compare with Remark 6.7.

Of course TestL∞(TM ) ⊂ Test(TM ),TestL∞(TM ) ⊂ L1(TM ) ∩ L∞(TM ), whilemerely Reg(TM ) ⊂ L2(TM ) ∩ L∞(TM ). By Lemma 3.15, we have Test(TM ) ⊂DTV(div) ∩ D(div) — as well as nX = 0 for every X ∈ Test(TM ) — while onlyReg(TM ) ⊂ D(div). By Lemma 2.7 and Proposition 1.24, all classes are dense inLp(TM ), p ∈ [1,∞), and weakly∗ dense in L∞(TM ). Furthermore, we set

TestL∞(TM ) := TestL∞(TM ),

Test(T ∗M ) := Test(TM ),

Reg(T ∗M ) := Reg(TM ).

4.3.4. Lebesgue spaces on tensor products. Denote the two-fold tensor products ofL2(T ∗M ) and L2(TM ), respectively, in the sense of Subsection 1.4.5 by

L2((T ∗)⊗2M ) := L2(T ∗M )⊗2,

L2(T⊗2M ) := L2(TM )⊗2.

By the discussion from Subsection 1.4.5, Theorem 2.4 and Proposition 1.24, bothare separable Hilbert modules. They are pointwise isometrically module isomorphic:the respective pairing is initially defined by

(ω1 ⊗ ω2)(X1 ⊗X2) := ω1(X1)ω2(X2) m-a.e.

for ω1, ω2 ∈ L2(T ∗M )∩L∞(T ∗M ) and X1, X2 ∈ L2(T ∗M )∩L∞(T ∗M ), and is exten-ded by linearity and continuity to L2((T ∗)⊗2M ) and L2(T⊗2M ), respectively. By

46 MATHIAS BRAUN

a slight abuse of notation, this pairing, with Proposition 1.24, induces the musicalisomorphisms : L2(T⊗2M ) → L2((T ∗)⊗2M ) and ♯ := −1 given by

A♯ : T := A(T ) =: A : T m-a.e.

We let Lp((T ∗)⊗2M ) and Lp(T⊗2M ), p ∈ 0 ∪ [1,∞], be defined similarlyto Subsection 4.3.1 and Subsection 4.3.2. For p ∈ [1,∞], these spaces naturallybecome Banach which, if p < ∞, are separable.

Lastly, we define the subsequent Lp-dense sets, p ∈ [1,∞], intended strongly ifp < ∞ and weakly∗ if p = ∞, reminiscent of Subsection 1.4.5:

TestL∞((T ∗)⊗2M ) := TestL∞(T ∗M )⊙2,

TestL∞(T⊗2M ) := TestL∞(TM )⊙2,

Test((T ∗)⊗2M ) := Test(T ∗M )⊙2,

Test(T⊗2M ) := Test(TM )⊙2,

Reg((T ∗)⊗2M ) := Reg(T ∗M )⊙2,

Reg(T⊗2M ) := Reg(TM )⊙2.

4.3.5. Lebesgue spaces on exterior products. Given any k ∈ N0, we set

L2(ΛkT ∗M ) := ΛkL2(T ∗M ),

L2(ΛkTM ) := ΛkL2(TM ),

where the exterior products are intended as in Subsection 1.4.7. For k ∈ 0, 1, weemploy the consistent interpretations

L2(Λ1T ∗M ) := L2(T ∗M ),

L2(Λ1TM ) := L2(TM ),

L2(Λ0T ∗M ) := L2(Λ0TM ) := L2(M ).

By Subsection 1.4.7, these are naturally Hilbert modules. As in Subsection 4.3.4,L2(ΛkT ∗M ) and L2(ΛkTM ) are pointwise isometrically module isomorphic. Forbrevity, the induced pointwise pairing between ω ∈ L2(ΛkT ∗M ) and X1 ∧ . . . Xk ∈L2(ΛkTM ), X1, . . . , Xk ∈ L2(TM ) ∩ L∞(TM ), is written

ω(X1, . . . , Xk) := ω(X1 ∧ · · · ∧Xk).

We let Lp(ΛkT ∗M ) and Lp(ΛkTM ), p ∈ 0 ∪ [1,∞], be as in Subsection 4.3.1and Subsection 4.3.2. For p ∈ [1,∞], these spaces are Banach and, if p < ∞,additionally separable.

Let the formal k-th exterior products, k ∈ N0, of the classes from Subsection 4.3.3be defined through

TestL∞(ΛkT ∗M ) := n∑

i=1

f0i df1

i ∧ · · · ∧ dfki : n ∈ N, f ji ∈ TestL∞(M ),

TestL∞(ΛkTM ) := n∑

i=1

f0i ∇f1

i ∧ · · · ∧ ∇fki : n ∈ N, f ji ∈ TestL∞(M ),

Test(ΛkT ∗M ) := n∑

i=1

f0i df1

i ∧ · · · ∧ dfki : n ∈ N, f ji ∈ Test(M ),

Test(ΛkTM ) := n∑

i=1

f0i ∇f1

i ∧ · · · ∧ ∇fki : n ∈ N, f ji ∈ Test(M )


Reg(ΛkT ∗M ) := n∑

i=1

f0i df1

i ∧ · · · ∧ dfki : n ∈ N, f ji ∈ Test(M ),

f0i ∈ Test(M ) ∪ R 1M

,n∑

i

Reg(ΛkTM ) := n∑

i=1

f0i ∇f1

i ∧ · · · ∧ ∇fki : n ∈ N, f ji ∈ Test(M ),

f0i ∈ Test(M ) ∪ R 1M

.

n∑

i

We employ the evident interpretations for k = 1, while the respective spaces fork = 0 are identified with those spaces to which their generic elements’s zeroth orderterms belong to. These classes are dense in their respective Lp-spaces, p ∈ [1,∞]

— strongly if p < ∞, and weakly∗ if p = ∞.

5. Hessian

5.1. The Sobolev space D(Hess). Now we define the key object of our secondorder differential structure, namely the Hessian of suitable functions f ∈ F. Wechoose an integration by parts procedure as in [45, Subsec. 3.3.1], motivated by thesubsequent Riemannian example.

Example 5.1. Let M be a Riemannian manifold with boundary. The metric compat-ibility of ∇ allows us to rephrase the definition of the Hessian Hess f ∈ Γ((T ∗)⊗2M )of a function f ∈ C∞(M ) pointwise as

2 Hess f(∇g1,∇g2) = 2⟨∇∇g1 ∇f,∇g2

⟩

=⟨∇g1,∇〈∇f,∇g2〉

⟩+

⟨∇g2,∇〈∇f,∇g1〉

⟩

−⟨∇f,∇〈∇g1,∇g2〉

⟩(5.1)

for every g1, g2 ∈ C∞c (M ), see e.g. [84, p. 28]. The first equality ensures that Hess f

is C∞-linear in both components. Thus, since smooth gradient vector fields locallygenerate TM , the second equality characterizes the Hessian of f .

We now restrict our attention to those g1 and g2 whose derivatives constituteNeumann vector fields, i.e.

〈∇g1, n〉 = 〈∇g2, n〉 = 0 on ∂M , (5.2)

e.g. to g1, g2 ∈ C∞c (M ). Multiply (5.1) by a function h ∈ C∞

c (M ) and integrate(by parts). In this case, recall that h∇g1, h∇g2 ∈ DTV(div) ∩ D(div) with

div(h∇g1) = divv(h∇g1) v,

div(h∇g2) = divv(h∇g2) v(5.3)

by Example 3.9. The resulting integral identity reads

2ˆ

M

hHess f(∇g1,∇g2) dv

= −ˆ

M

〈∇f,∇g2〉 divv(h∇g1) dv −ˆ

M

〈∇f,∇g1〉 divv(h∇g2) dv

−ˆ

M

h⟨∇f,∇〈∇g1,∇g2〉

⟩dv.

In turn, this integral identity characterizes Hess f on M by the arbitrariness of g1,g2 and h, and hence on M by the existence of a smooth extension to all of M .

48 MATHIAS BRAUN

Observe that on the r.h.s. of the previous integral identity, no second orderexpression in f is present. Moreover, as we have already noted in Lemma 3.15,Test(TM ) is a large class of vector fields obeying a nonsmooth version of (5.2).Next, note that all volume integrals on the r.h.s. — with v replaced by m — are well-defined for every f ∈ F and g1, g2, h ∈ Test(M ). Indeed, the first one exists since∇g2 ∈ L∞(M ), whence 〈∇f,∇g2〉 ∈ L2(M ), and since h∇g1 ∈ DTV(div) ∩ D(div)with div(h∇g1) ∈ L2(M ) by Lemma 3.15. An analogous argument applies for thesecond volume integral. The last one is well-defined since 〈∇g1,∇g2〉 ∈ F thanksto Proposition 4.20 and polarization, and since h ∈ L∞(M ).

These observations lead to the following definition.

Definition 5.2. We define the space D(Hess) to consist of all f ∈ F for whichthere exists A ∈ L2((T ∗)⊗2M ) such that for every g1, g2, h ∈ Test(M ),

2ˆ

M

hA(∇g1,∇g2) dm

= −ˆ

M

〈∇f,∇g1〉 div(h∇g2) dm −ˆ

M

〈∇f,∇g2〉 div(h∇g1) dm

−ˆ

M

h⟨∇f,∇〈∇g1,∇g2〉

⟩dm.

If such an A exists, it is unique, denoted by Hess f and termed the Hessian of f .

Indeed, given any f ∈ D(Hess) there is at most one A as in Definition 5.2 bydensity of Test(T⊗2M ) in L2(T⊗2M ), since Test(M ) is an algebra. In particular,D(Hess) is a vector space and Hess is a linear operator on it. Further elementaryproperties are collected in Theorem 5.3.

The space D(Hess) is endowed with the norm ‖ · ‖D(Hess) given by∥∥f

∥∥2

D(Hess):=

∥∥f∥∥2

L2(M)+

∥∥df∥∥2

L2(T∗M)+

∥∥Hess f∥∥2

L2((T∗)⊗2M).

Furthermore we define the energy functional E2 : F→ [0,∞] by

E2(f) :=

ˆ

M

∣∣Hess f∣∣2

HSdm if f ∈ D(Hess),

∞ otherwise.

Theorem 5.3. The space D(Hess), the Hessian Hess and the functional E2 havethe following properties.

(i) D(Hess) is a separable Hilbert space w.r.t. ‖ · ‖D(Hess).(ii) The Hessian is a closed operator on D(Hess), i.e. the image of the map

Id × Hess : D(Hess) → F × L2((T ∗)⊗2M ) is a closed subspace of F ×L2((T ∗)⊗2M ).

(iii) For every f ∈ D(Hess), the tensor Hess f is symmetric, i.e.

Hess f = (Hess f)⊤

according to the definition of the transpose from (1.10).(iv) E2 is F-lower semicontinuous, and for every f ∈ F,

E2(f) = sup

−r∑

l=1

ˆ

M

〈∇f,∇gl〉 div(hl h′l ∇g′

l) dm

−r∑

l=1

ˆ

M

〈∇f,∇g′l〉 div(hl h′

l ∇gl) dm

−r∑

l=1

ˆ

M

hl h′l

⟨∇f,∇〈∇gl,∇g′

l〉⟩

dm


−ˆ

M

∣∣∣r∑

l=1

hl h′l ∇gl ⊗ ∇g′

l

∣∣∣2

dm :

r ∈ N, gr, g′r, hr, h

′r ∈ Test(M )

.

Proof. Item (ii) follows since the r.h.s. of the defining property of the Hessian inDefinition 5.2 is continuous in f andA w.r.t. weak convergence in Fand L2((T ∗)⊗2M ),respectively, for fixed g1, g2, h ∈ Test(M ).

The Hilbert space property of D(Hess) in (i) is a direct consequence of the com-pleteness of F, (ii) and since ‖ · ‖D(Hess) trivially satisfies the parallelogram identity.Hence, we are left with the separability of D(Hess). Since F and L2((T ∗)⊗2M ) areseparable by Proposition 1.5, Lemma 2.7 and the discussion from Subsection 1.4.5,their Cartesian product is a separable Hilbert space w.r.t. the norm ‖ · ‖, where

‖(f,A)‖2 :=∥∥f

∥∥2

F+

∥∥A∥∥2

L2((T∗)⊗2M).

In particular, Id × Hess : D(Hess) → F× L2((T ∗)⊗2M ) is a bijective isometry ontoits image, whence the claim follows from (ii).

Concerning (iii), setting h := h1 h2 with h1, h2 ∈ Test(M ), we easily see that forevery g1, g2 ∈ Test(M ) the r.h.s. of the defining property of Hess f , f ∈ D(Hess),in Definition 5.2 is symmetric in h1 and h2 as well as g1 and g2, respectively, andit is furthermore bilinear in h1 ∇g1 and h2 ∇g2. Hence, using (1.10) we deduce thesymmetry of Hess f , f ∈ D(Hess), on Test(T⊗2M ) and hence on all of L2(T⊗2M )by a density argument.

The F-lower semicontinuity of E2 in (iv) directly follows since bounded subsetsof the Hilbert space L2((T ∗)⊗2M ) are weakly relatively compact, combined withMazur’s lemma and (ii).

We finally turn to the duality formula in (iv).Let us first prove the inequality “≥”, for which we assume without restriction

that f ∈ D(Hess). By duality of D(Hess) and its Hilbert space dual D(Hess)′ aswell as the density of Test(T⊗2M ) in L2(T⊗2M ),

E2(f) = sup

2n∑

i=1

ˆ

M

Hess f(Xi, X′i) dm

−ˆ

M

∣∣∣n∑

i=1

Xi ⊗X ′i

∣∣∣2

dm : n ∈ N, Xi, X′i ∈ Test(TM )

.

Let us write Xi ⊗X ′i := hi1 h

′i1 ∇gi1 ⊗ ∇g′

i1 + · · · + him h′im ∇gim ⊗ g′

im for certainelements gij , g′

ij , hij , h′ij ∈ Test(M ), i ∈ 1, . . . , n and j ∈ 1, . . . ,m with n,m ∈

N. Then by Definition 5.2 and since Test(M ) is an algebra,

2n∑

i=1

ˆ

M

Hess f(Xi, X′i) dm

= 2n∑

i=1

m∑

j=1

ˆ

M

hi1 h′i1 Hess f(∇gi1,∇g′

i1) dm

= −n∑

i=1

m∑

j=1

ˆ

M

〈∇f,∇gij〉 div(hij h′ij ∇g′

ij) dm

−n∑

i=1

m∑

j=1

ˆ

M

〈∇f,∇g′ij〉 div(hij h′

ij ∇gij) dm

−n∑

i=1

m∑

j=1

ˆ

M

hij h′ij

⟨∇f,∇〈∇gij ,∇g′

ij〉⟩

dm,

50 MATHIAS BRAUN

which terminates the proof of “≥”.Turning to “≤” in (iv), we may and will assume without loss of generality that

the supremum, henceforth denoted by C, on the r.h.s. of the claimed formula isfinite. Consider the operator Φ: Test(T⊗2M ) → R given by

2 Φn∑

i=1

m∑

j=1

hij h′ij ∇gij ⊗ ∇g′

ij

:= −n∑

i=1

m∑

j=1

ˆ

M

〈∇f,∇gij〉 div(hij h′ij ∇g′

ij) dm

−n∑

i=1

m∑

j=1

ˆ

M

〈∇f,∇g′ij〉 div(hij h′

ij ∇gij) dm

−n∑

i=1

m∑

j=1

ˆ

M

hij h′ij

⟨∇f,∇〈∇gij ,∇g′

ij〉⟩

dm.

(5.4)

The value of Φ(T ) is independent of the particular way of writing T ∈ Test(T⊗2M ).Indeed, if T = 0 but Φ(T ) 6= 0, letting λ → ∞ in the identity Φ(λT ) sgn Φ(T ) =Φ(T )λ sgn Φ(T ) implied by (5.4) would contradict the assumption that C < ∞.The map Φ is thus well-defined, it is linear, and for every T ∈ Test(T⊗2M ),

2 Φ(T ) ≤ C +∥∥T

∥∥2

L2(T⊗2M).

Replacing T by λT and optimizing over λ ∈ R gives

|Φ(T )| ≤√C ‖T ‖L2(T⊗2M) (5.5)

for every T ∈ Test(T⊗2M ). Hence, Φ uniquely induces a (non-relabeled) elementof the Hilbert space dual L2(T⊗2M )′ of L2(T⊗2M ). By Proposition 1.24, we finda unique element A′ ∈ L2((T ∗)⊗2M ) such that

Φ(T ) =ˆ

M

A′(T ) dm

for every T ∈ L2(T⊗2M ). Now Lemma 4.17, Lemma 3.15 as well as the continuityof Φ allow us to replace the terms hij h′

ij by arbitrary elements kij ∈ Test(M ),still retaining the identity (5.4) with kij in place of hij h′

ij , i ∈ 1, . . . , n andj ∈ 1, . . . ,m. In particular, by Definition 5.2, we deduce that f ∈ D(Hess) andA′ = Hess f . By Proposition 1.24 again and (5.5), we obtain

‖Hess f‖L2((T∗)⊗2M) = ‖Φ‖L2(T⊗2M)′ ≤√C,

which is precisely what was left prove.

Remark 5.4. If E2 is extended to L2(M ) by E2(f) := ∞ for f ∈ L2(M ) \ F, itis unclear if the resulting functional is L2-lower semicontinuous. To bypass thisissue in applications, one might instead use that by Theorem 5.3, the functionalEε2 : L2(M ) → [0,∞] given by

Eε2(f) :=

ε

ˆ

M

|∇f |2 dm +ˆ

M

∣∣Hess f∣∣2

HSdm if f ∈ D(Hess),

∞ otherwise

is L2-lower semicontinuous for every ε > 0.If M is, say, a compact Riemannian manifold without boundary, one can easily

prove using the Bochner identity that the (nonpositive) generator associated withEε2 [43, Thm. 1.3.1] is the Paneitz-type operator

−∆2 + ε∆f + div(Ric ∇·).


Remark 5.5. In general, the Hessian is not the trace of the Laplacian in the sense of(1.14). This already happens on weighted Riemannian manifolds without boundary:of course, the associated Laplacian ∆ is defined by partial integration w.r.t. thereference measure [48, Sec. 3.6], while the definition of Hessian only depends onthe metric tensor. See also the second part of Example 3.9. Examples of abstractspaces for which this is the case — and which currently enjoy high research interest[21, 34, 66] — are noncollapsed RCD(K,N) spaces, K ∈ R and N ∈ [1,∞) [34,Thm. 1.12]. See also Remark 5.16 below.

Remark 5.6. In line with Remark 5.5, although a priori m plays a role in Definition5.2, we expect the Hessian to only depend on conformal transformations of 〈·, ·〉,but not on drift transformations of m. For instance, this is known on RCD∗(K,N)spaces, K ∈ R and N ∈ [1,∞), see e.g. [54, Prop. 3.11] or [56, Lem. 2.16], and itdoes not seem hard to adapt the arguments from [54] to more general settings.

Remark 5.7. As an alternative to Definition 5.2, one can define D(Hess) as thefiniteness domain of the r.h.s. of the duality formula in (iv) in Theorem 5.11. TheHessian of f ∈ D(Hess) is then well-defined by the same duality arguments as inthe proof of Theorem 5.11.

5.2. Existence of many functions in D(Hess). Up to now, we still do not knowwhether D(Hess) is nonempty. The ultimate goal of this section is to prove thatTest(M ) ⊂ D(Hess) in Theorem 5.11, whence D(Hess) is even dense in L2(M ).

The strategy is reliant on [45, Subsec. 3.3.2], which has itself been inspired bythe “self-improvement” works [8, 9], see [45, Rem. 3.3.10] and also [41, 87, 96]. Thekey technical part (not only for Theorem 5.11, but in fact for Theorem 8.9 below aswell) is contained in Lemma 5.9, where — loosely speaking and up to introducingthe relevant objects later — we show that

|∇X : T |2 ≤[∆2κ |X |2

2+

⟨X, (~∆X)♯

⟩−

∣∣(∇X)asym

∣∣2

HS

] ∣∣T∣∣2

HSm-a.e.

for X,T ∈ Test(TM ). Of course, neither we introduced the covariant derivative∇, Definition 6.2 or the Hodge Laplacian ~∆, Definition 7.21, yet, nor in general wehave |X |2 ∈ D(∆2κ) for X ∈ Test(TM ). Reminiscent of Proposition 4.20 and [41,Cor. 6.3], we instead rephrase the above inequality in terms of measures, and theinvolved objects ∇X and ~∆X therein as the “r.h.s.’s of the identities one wouldexpect for ∇X and ~∆X for X ∈ Test(M )”, rigorously proven in Theorem 6.3 andLemma 8.1 below. In particular, by optimization over T ∈ Test(TM ),

∣∣(∇X)sym

∣∣2

HS≤ ∆2κ |X |2

2+

⟨X, (~∆X)♯

⟩−

∣∣(∇X)asym

∣∣2

HSm-a.e.,

which is the Bochner inequality for vector fields according to (1.11). For X := ∇f ,f ∈ Test(M ), this essentially provides Theorem 5.11. Details about this inequalityfor general X ∈ Reg(TM ), leading to Theorem 8.9, are due to Lemma 8.2.

We start with a technical preparation. Given µ, ν ∈ M+f (M ), we define the

Borel measure√µ ν ∈ M

+f (M ) as follows. Let ι ∈ M

+f (M ) with µ ≪ ι and ν ≪ ι

be arbitrary, denote the respective densities w.r.t. ι by f, g ∈ L1(M , ι), and set√µ ν :=

√f g ι.

For instance, one can choose ι := |µ| + |ν| [52, Thm. 30.A] — in fact, the previousdefinition is independent of the choice of ι, whence

√µ ν is well-defined.

The following important measure theoretic lemma is due to [45, Lem. 3.3.6].

Lemma 5.8. Let µ1, µ2, µ3 ∈ M±f (M ) satisfy the inequality

λ2 µ1 + 2λµ2 + µ3 ≥ 0

52 MATHIAS BRAUN

for every λ ∈ R. Then the following properties hold.

(i) The elements µ1 and µ3 are nonnegative, and

|µ2| ≤ √µ1 µ3.

(ii) We have µ2 ≪ µ1, µ2 ≪ µ3 and

‖µ2‖TV ≤√

‖µ1‖TV ‖µ3‖TV.

(iii) The m-singular parts (µ1)⊥ and (µ3)⊥ of µ1 and µ3 are nonnegative. More-over, expressing the densities of the m-absolutely continuous parts of µi byρi := d(µi)≪/dm ∈ L1(M ), i ∈ 1, 2, 3, we have

|ρ2|2 ≤ ρ1 ρ3 m-a.e.

In the subsequent lemma, all terms where N ′ is infinite are interpreted as beingzero. Similar proofs can be found in [16, 45, 53].

Lemma 5.9. Let N ′ ∈ [N,∞], n,m ∈ N, f, g ∈ Test(M )n and h ∈ Test(M )m.Define µ1[f, g] ∈ M

±f (M ) as

µ1[f, g] :=n∑

i,i′=1

gi gi′ Γ2κ2 (fi, fi′) + 2

n∑

i,i′=1

gi H[fi](fi′ , gi′)m

+12

n∑

i,i′=1

[〈∇fi,∇fi′〉〈∇gi,∇gi′〉 + 〈∇fi,∇gi′〉〈∇gi,∇fi′〉

]m

− 1N ′

[ n∑

i=1

[gi ∆fi + 〈∇fi,∇gi〉

]]2

m

As in Lemma 5.8, we denote the density of the m-absolutely continuous part ofµ1[f, g] by ρ1[f, g] := dµ1[f, g]≪/dm ∈ L1(M ). Then the m-singular part µ1[f, g]⊥of µ1[f, g] as well as ρ1[f, g] are nonnegative, and

[ n∑

i=1

m∑

j=1

[〈∇fi,∇hj〉〈∇gi,∇hj〉 + gi H[fi](hj , hj)

]

− 1N ′

n∑

i=1

m∑

j=1


]|∇hj |2

]2

≤ ρ1[f, g][ m∑

j,j′=1

〈∇hj ,∇hj′ 〉2 − 1N ′

[ m∑

j=1

|∇hj |2]2]

m-a.e.

Proof. We define µ2[f, g, h], µ3[h] ∈ M±f (M ) by

µ2[f, g, h] :=n∑

i=1

m∑

j=1

[〈∇fi,∇hj〉〈∇gi,∇hj〉 + gi H[fi](hj , hj)

]m

− 1N ′

n∑

i=1

m∑

j=1


]|∇hj |2 m,

µ3[h] :=[ m∑

j,j′=1

〈∇hj ,∇hj′〉2 − 1N ′

[ m∑

j=1

|∇hj |2]2]

m.

Both claims readily follow from Lemma 5.8 as soon as λ2 µ1[f, g] + 2λµ2[f, g, h] +µ3[h] ≥ 0 for every λ ∈ R, which is what we concentrate on in the sequel.


Let λ ∈ R and pick a, b ∈ Rn as well as c ∈ Rm. Define the function ϕ ∈C∞(R2n+m) through

ϕ(x, y, z) :=n∑

i=1

[λxi yi + ai xi − bi yi

]+

m∑

j=1

[(zj − cj)2 − c2

j

].

For every i ∈ 1, . . . , n and every j ∈ 1, . . . ,m, those first and second partialderivatives of ϕ which, do not always vanish identically read

ϕi(x, y, z) = λ yi + ai,

ϕn+i(x, y, z) = λxi − bi,

ϕ2n+j(x, y, z) = 2(zj − cj),

ϕi,n+i(x, y, z) = λ,

ϕn+i,i(x, y, z) = λ,

ϕ2n+j,2n+j(x, y, z) = 2.

For convenience, we write

A2κ(λ, a, b, c) := A2κ[ϕ q],B(λ, a, b, c) := B[ϕ q],C(λ, a, b, c) := C[ϕ q],D(λ, a, b, c) := D[ϕ q],

where the respective r.h.s.’s are defined as in Lemma 4.21 for α := 2n + m andq := (f, g, h). Using the same Lemma 4.21, we compute

A2κ(λ, a, b, c) =n∑

i,i′=1

(λ gi + ai) (λ gi′ + ai′) Γ2κ2 (fi, fi′) + other terms,

B(λ, a, b, c) = 4n∑

i,i′=1

(λ gi + ai)λH[fi](fi′ , gi′)

+ 4n∑

i=1

m∑

j=1

(λ gi + ai) H[fi](hj , hj) + other terms,

C(λ, a, b, c) = 2n∑

i,i′=1

λ2[〈∇fi,∇fi′〉〈∇gi,∇gi′〉 + 〈∇fi,∇gi′〉〈∇gi,∇fi′〉

]

+ 8n∑

i=1

m∑

j=1

λ 〈∇fi,∇hj〉〈∇gi,∇hj〉

+ 4m∑

j,j′=1

〈∇hj ,∇hj′ 〉2 + other terms,

D(λ, a, b, c) =n∑

i,i′=1

(λ gi + ai) (λ gi′ + ai) ∆fi ∆fi′

+ 4n∑

i,i′=1

λ (λ gi + ai) ∆fi 〈∇fi′ ,∇gi′〉

+ 4n∑

i,i′=1

λ2 〈∇fi,∇gi〉〈∇fi′ ,∇gi′〉

54 MATHIAS BRAUN

+ 4n∑

i=1

m∑

j=1

(λ gi + ai) ∆fi |∇hj |2

+ 8n∑

i=1

m∑

j=1

λ 〈∇fi,∇gi〉 |∇hj |2

+ 4[ m∑

j=1

|∇hj |2]2

+ other terms.

Here, every “other term” contains at least one factor of the form λ fi − bi or hj − cjfor some i ∈ 1, . . . , n and j ∈ 1, . . . ,m.

By Lemma 4.21 and Proposition 4.20 with the nonnegativity of D(λ, a, b, c) aswell as the trivial inequality 1/N ≥ 1/N ′,

A2κ(λ, a, b, c) +[B(λ, a, b, c) + C(λ, a, b, c) − 1

N ′D(λ, a, b, c)

]m ≥ 0.

By the arbitrariness of a, b ∈ Rn and c ∈ Rm, for every Borel partition (Ep)p∈N ofM , every Borel set F ⊂ M and all sequences (ak)k∈N and (bk)k∈N in Rn as well as(ck)k∈N in Rm,

1F∑

k∈N

1Ek

[A2κ(λ, ak, bk, ck) +

[B(λ, ak, bk, ck) + C(λ, ak, bk, ck)

− 1N ′

D(λ, ak, bk, ck)]m

]≥ 0.

(5.6)

We now choose the involved quantities appropriately. Let (Fk)k∈N be an E-nestwith the property that the restrictions of f , g and h to Fk are continuous for everyk ∈ N, and set F :=

⋃k∈N

Fk. Since F c is an E-polar set and thus not seen by m

and Γ2κ2 (fi, fi′), i, i′ ∈ 1, . . . , n, its contribution to the subsequent manipulations

is ignored. For l ∈ N we now take a Borel partition (Elk)k∈N of M and sequences(alk)k∈N and (blk)k∈N in Rn as well as (clk)k∈N in Rm with

supk,l∈N

[|alk| + |blk| + |clk|

]< ∞

in such a way that

liml→∞

∑

k∈N

1Elkalk = λ g,

liml→∞

∑

k∈N

1Elkblk = λ f ,

liml→∞

∑

k∈N

1Elkclk = h

pointwise on F . Thus, the l.h.s. of (5.6) with (Ek)k∈N, (ak)k∈N, (bk)k∈N and(ck)k∈N replaced by (Elk)k∈N, (alk)k∈N, (blk)k∈N and (clk)k∈N, l ∈ N, respectively,converges w.r.t. ‖ · ‖TV as l → ∞. In fact, in the limit as l → ∞ every “other term”above becomes zero, and the prefactors λ gi + (alk)i become 2λ gi, i ∈ 1, . . . , n.We finally obtain

4λ2n∑

i,i′=1

gi gi′ Γ2κ2 (fi, fi′)

+ 8λ2n∑

i,i′=1

gi H[fi](fi′ , gi′)m + 8λn∑

i=1

m∑

j=1

gi H[fi](hj , hj)m


+ 2λ2n∑

i,i′=1

[〈∇fi,∇fi′〉〈∇gi,∇gi′〉 + 〈∇fi,∇gi′〉〈∇gi,∇fi′〉

]m

+ 8λn∑

i=1

m∑

j=1

〈∇fi,∇hj〉〈∇gi,∇hj〉m

+ 4m∑

j,j′=1

〈∇hj ,∇hj′〉2m

− 4λ2

N ′

[ n∑

i=1


]]2

m

− 8λN ′

n∑

i=1

m∑

j=1


]|∇hj |2 m

− 4N ′

[ m∑

j=1

|∇hj |2]2

≥ 0.

Dividing by 4 and sorting terms by the order of λ yields the claim.

We note the following consequence of Lemma 5.9 that is used in Theorem 5.11below as well, but becomes especially important in Section 5.3.

Remark 5.10. The nonnegativity of µ3[h] from Lemma 5.9 can be translated intothe following trace inequality, compare with [16, Rem. 2.19] and the proof of [53,Prop. 3.2]. With the pointwise trace defined as in (1.14), we have

∣∣∣m∑

j=1

∇hj ⊗ ∇hj∣∣∣2

HS≥ 1N

tr[ m∑

j=1

∇hj ⊗ ∇hj]2

m-a.e.

for every m ∈ N and every h ∈ Test(M )m.

Theorem 5.11. Every f ∈ Test(M ) belongs to D(Hess) and satisfies

Hess f(∇g1,∇g2) = H[f ](g1, g2) m-a.e. (5.7)

for every g1, g2 ∈ Test(M ). Moreover, denoting by γ2κ2 (f) ∈ L1(M ) the density of

the m-absolutely continuous part of Γ2κ2 (f), we have

∣∣Hess f∣∣2

HS≤ γ2κ

2 (f) m-a.e. (5.8)

Proof. Recall that indeed γ2κ2 (f) ∈ L1(M ) by Proposition 4.24. Let g, h1, . . . , hm ∈

Test(M ), m ∈ N. Applying Lemma 5.9 for N ′ := ∞ and n := 1 then entails

[ m∑

j=1

[〈∇f,∇hj〉〈∇g,∇hj〉 + gH[f ](hj , hj)

]]2

≤[g2 γ2κ

2 (f) + 2gH[f ](f, g) +12

|∇f |2 |∇g|2 +12

〈∇f,∇g〉2]

×m∑

j,j′=1

〈∇hj ,∇hj′ 〉2

=[g2 γ2κ

2 (f) + g⟨∇|∇f |2,∇g

⟩+

12

|∇f |2 |∇g|2 +12

〈∇f,∇g〉2]

×m∑

j,j′=1

〈∇hj ,∇hj′ 〉2m-a.e.

56 MATHIAS BRAUN

In the last identity, we used the definition (4.4) of H[f ](f, g). Using the first partof Lemma 4.18 and possibly passing to subsequences, this m-a.e. inequality ex-tends to all g ∈ F∩ L∞(M ). Thus, successively setting g := gn, n ∈ N, where(gn)n∈N is the sequence provided by Lemma 1.9, together with the locality of ∇from Proposition 2.11, and by the definition (1.9) of the pointwise Hilbert–Schmidtnorm of L2(T⊗2M ), we obtain

∣∣∣m∑

j=1

H[f ](hj , hj)∣∣∣ ≤ γ2κ

2 (f)1/2∣∣∣m∑

j=1

∇hj ⊗ ∇hj∣∣∣HS

m-a.e. (5.9)

This implies pointwise m-a.e. off-diagonal estimates as follows. Given any m′ ∈ N

and hj , h′j ∈ Test(M ), j ∈ 1, . . . ,m′, since

m′∑

j=1

H[f ](hj , h′j) =

12

m′∑

j=1

[H[f ](hj + h′

j , hj + h′j) − H[f ](hj , hj) − H[f ](h′

j , h′j)

]

holds m-a.e., applying (5.9), using that

12

m′∑

j=1

[∇(hj + h′

j) ⊗ ∇(hj + h′j) − ∇hj ⊗ ∇hj − ∇h′

j ⊗ ∇h′j

]

=12

m′∑

j=1

[∇hj ⊗ ∇h′

j + ∇h′j ⊗ ∇hj

]

=[ m′∑

j=1

∇hj ⊗ ∇h′j

]sym

and finally employing that |Tsym|HS ≤ |T |HS for every T ∈ L2(T⊗2M ), we get

∣∣∣m′∑

j=1

H[f ](hj , h′j)

∣∣∣ ≤ γ2κ2 (f)1/2

∣∣∣12

m′∑

j=1

[∇hj ⊗ ∇h′

j + ∇h′j ⊗ ∇hj

]∣∣∣HS

≤ γ2κ2 (f)1/2

∣∣∣m′∑

j=1

∇hj ⊗ ∇h′j

∣∣∣HS

m-a.e.

We replace hj by aj hj , j ∈ 1, . . . ,m′, for arbitrary a1, . . . , am′ ∈ Q. This gives

∣∣∣m′∑

j=1

aj H[f ](hj , h′j)

∣∣∣ ≤ γ2κ2 (f)1/2

∣∣∣m′∑

j=1

aj ∇hj ⊗ ∇h′j

∣∣∣HS

m-a.e. (5.10)

In fact, since Q is countable, we find an m-negligible Borel set B ⊂ M on whosecomplement (5.10) holds pointwise for every a1, . . . , am′ ∈ Q. Since both sides of(5.10) are continuous in a1, . . . , am′ , by density of Q in R we deduce that (5.10)holds pointwise on Bc for every a1, . . . , am′ ∈ R. Therefore, given any g1, . . . , gm′ ∈Test(M ), up to possibly removing a further m-negligible Borel set C ⊂ M , for everyx ∈ (B ∪ C)c we may replace aj by gj(x), j ∈ 1, . . . ,m′, in (5.10). This leads to

∣∣∣m′∑

j=1

gj H[f ](hj , h′j)

∣∣∣2

≤ γ2κ2 (f)1/2

∣∣∣m′∑

j=1

gj ∇hj ⊗ ∇h′j

∣∣∣HS

m-a.e. (5.11)

We now define the operator Φ: Test(T⊗2M ) → L0(M ) by

Φm′∑

j=1

gj g′j ∇hj ⊗ ∇h′

j :=m′∑

j=1

gj g′j H[f ](hj , h′

j). (5.12)


From (5.11) and the algebra property of Test(M ), it follows that Φ is well-defined,i.e. the value of Φ(T ) does not depend on the specific way of representing a givenelement T ∈ Test(T⊗2M ). Moreover, the map Φ is clearly linear, and for everyg ∈ Test(M ) and every T ∈ Test(T⊗2M ),

Φ(g T ) = gΦ(T ). (5.13)

Since the m-singular part Γ2κ2 (f)⊥ of Γ2κ

2 (f) is nonnegative, by (4.5) we getˆ

M

γ2κ2 (f) dm ≤ Γ2κ

2 (f)[M ] =ˆ

M

(∆f)2 dm −⟨κ

∣∣ |∇f |2⟩. (5.14)

After integrating (5.11) and employing Cauchy–Schwarz’s inequality,

‖Φ(T )‖L1(M) ≤[ˆ

M

(∆f)2 dm −⟨κ

∣∣ |∇f |2⟩]1/2

‖T ‖L2(T⊗2M)

holds for every T ∈ Test(T⊗2M ). Thus, by density of Test(T⊗2M ) in L2(T⊗2M )and (5.13), Φ uniquely extends to a (non-relabeled) continuous, L∞-linear mapfrom L2(T⊗2M ) into L1(M ), whence Φ ∈ L2((T ∗)⊗2M ) by definition of the latterspace.

To check that f ∈ D(Hess) and Φ = Hess f , first note that by the continuity of Φand Lemma 4.17, we can replace gj g′

j by arbitrary kj ∈ Test(M ), j ∈ 1, . . . ,m′,still retaining the identity (5.12). Therefore, slightly changing the notation in (5.12),let g1, g2, h ∈ Test(M ) and use (5.13), the definition (4.4) of H[f ] and Lemma 3.15to derive that

2ˆ

M

hΦ(∇g1,∇g2) dm

=ˆ

M

h⟨∇g1,∇〈∇f,∇g2〉

⟩dm +

ˆ

M

h⟨∇g1,∇〈∇f,∇g2〉

⟩dm

−ˆ

M

h⟨∇g1,∇〈∇f,∇g2〉

⟩dm

= −ˆ

M


M


−ˆ

M

h⟨∇g1,∇〈∇f,∇g2〉

⟩dm,

which is the desired assertion f ∈ D(Hess) and Φ = Hess f .The same argument gives (5.7), while the inequality (5.8) is due to (5.11), the

density of Test(T⊗2M ) in L2(T⊗2M ) as well as the definition (1.9) of the pointwiseHilbert–Schmidt norm.

Theorem 5.11 implies the following qualitative result. A quantitative version ofit, as directly deduced in [45, Cor. 3.3.9] from [45, Thm. 3.3.8], is however not yetavailable only with the information collected so far. See Remark 5.13 below.

Corollary 5.12. Every f ∈ D(∆) belongs to the closure of Test(M ) in D(Hess),and in particular to D(Hess). More precisely, let ρ′ ∈ (0, 1) and α′ ∈ R be as inLemma 4.5 for µ := κ−. Then for every f ∈ D(∆), we have f ∈ D(Hess) with

ˆ

M

∣∣Hess f∣∣2

HSdm ≤ 1

1 − ρ′

ˆ

M

(∆f)2 dm +α′

1 − ρ′

ˆ

M

|∇f |2 dm.

Proof. Since (M ,E,m) satisfies BE1(κ,∞) by [41, Thm. 6.9], it also trivially obeysBE1(−κ−,∞), see also [41, Prop. 6.7]. As in the proof of Proposition 4.24,

E−κ−(

|∇f |)

≤ˆ

M

(∆f)2 dm

58 MATHIAS BRAUN

holds for every f ∈ Test(M ). Hence, using (4.1) we estimate⟨κ−

∣∣ |∇f |2⟩

≤ ρ′E(|∇f |

)+ α′

ˆ

M

|∇f |2 dm

= ρ′E−κ−(

|∇f |)

+ ρ′⟨κ−

∣∣ |∇f |2⟩

+ α′

ˆ

M

|∇f |2 dm

≤ ρ′

ˆ

M

(∆f)2 dm + ρ′⟨κ−

∣∣ |∇f |2⟩

+ α′

ˆ

M

|∇f |2 dm.

The claim for f ∈ Test(M ) now follows easily. We already know from Theorem 5.11that f ∈ D(Hess). Integrating (5.8) and using (5.14) thus yields

ˆ

M

∣∣Hess f∣∣2

HSdm ≤

ˆ

M

(∆f)2 dm −⟨κ

∣∣ |∇f |2⟩

(5.15)

≤ˆ

M

(∆f)2 dm +⟨κ−

∣∣ |∇f |2⟩

≤ 11 − ρ′

ˆ

M

(∆f)2 dm +α′

1 − ρ′

ˆ

M

|∇f |2 dm.

Finally, given f ∈ D(∆), let fn := maxminf, n,−n ∈ L2(M ) ∩ L∞(M ),n ∈ N. Note that ptfn ∈ Test(M ) for every t > 0 and every n ∈ N, and thatptfn → ptf in F as well as, thanks to Lemma 1.17, ∆ptfn → ∆ptf in L2(M ) asn → ∞. Moreover ptf → f in F as well as ∆ptf = pt∆f → ∆f in L2(M ) as t → 0.These observations imply that f belongs to the closure of Test(M ) in D(Hess),whence f ∈ D(Hess) by Theorem 5.3, and the claimed inequality, with unchangedconstants, is clearly stable under this approximation procedure.

Remark 5.13. The subtle reason why we still cannot deduce (5.15) for generalf ∈ D(∆) is that we neither know whether the r.h.s. of (5.15) makes sense —which essentially requires |∇f | ∈ F — nor, in the notation of the previous proof,whether

⟨κ

∣∣ |∇ptfn|2⟩

→⟨κ

∣∣ |∇f |2⟩

as n → ∞ and t → 0. (Neither we know ifE

(|∇ptfn|

)→ E

(|∇f |

)as n → ∞ and t → 0.) Both points are trivial in the more

restrictive RCD(K,∞) case from [45, Cor. 3.3.9], K ∈ R. In our setting, solelyLemma 4.5 does not seem sufficient to argue similarly. Instead, both points willfollow from Lemma 6.13 and Lemma 8.2, see Corollary 6.14 and Corollary 8.3.

5.3. Structural consequences of Lemma 5.9. Solely in this section, we assumethat BE2(κ,N) holds for N < ∞. In this case, we derive a nontrivial upper boundon the local dimension of L2(TM ) — and hence of L2(T ∗M ) by Proposition 1.24 —in Proposition 5.14. Our proof follows [53, Prop. 3.2]. Reminiscent of Remark 1.31and Corollary 2.8, as a byproduct we obtain an upper bound on the Hino index ofM . The key point is the trace inequality derived in Remark 5.10.

Let (En)n∈N∪∞ be the dimensional decomposition of L2(TM ) as provided byProposition 1.29. The maximal essential local dimension of L2(TM ) is

dimL∞,max L2(TM ) := supn ∈ N ∪ ∞ : m[En] > 0

.

Proposition 5.14. We have

dimL∞,max L2(TM ) ≤ ⌊N⌋.Moreover, if N is an integer, then for every f ∈ Test(M ),

tr Hess f = ∆f m-a.e. on EN .

Proof. Suppose to the contrapositive that m[En] > 0 for some n ∈ N ∪ ∞ withn > N . Let m ∈ N be a finite number satisfying N < m ≤ n. Let B ⊂ Em be agiven Borel set of finite, but positive m-measure. By Theorem 2.4 for µ := m and[45, Thm. 1.4.11], there exist vectors V1, . . . , Vm ∈ L2(TM ) such that 1Bc Vi = 0


and 〈Vi, Vj〉 = δij m-a.e. in B for every i, j ∈ 1, . . . ,m which generate L2(TM )on B. Recall that by Remark 5.10, for every h ∈ Test(M )m,

∣∣∣m∑

j=1

∇hj ⊗ ∇hj∣∣∣2

HS≥ 1N

tr[ m∑

j=1

∇hj ⊗ ∇hj]2

m-a.e. (5.16)

By a similar argument as for Theorem 5.11, we can replace ∇hj by fj ∇hj , j ∈1, . . . ,m, for arbitrary f1, . . . , fm ∈ L∞(M ), still retaining (5.16). By Lemma 2.7and Section 4.3, the linear span of such vector fields generates L2(TM ) on B. Wecan thus further replace fj ∇hj by 1B Vj , j ∈ 1, . . . ,m, and (5.16) translates into

m =∣∣∣m∑

j=1

Vj ⊗ Vj

∣∣∣2

HS≥ 1N

tr[ m∑

j=1

Vj ⊗ Vj

]2

=m2

Nm-a.e. on B. (5.17)

This is in contradiction with the assumption N < m.The second claim is only nontrivial if m[EN ] > 0. In this case, retain the notation

of the previous part and observe that under our given assumptions, equality occursin (5.17) for m replaced by N . Using Lemma 5.9, (5.7) and similar arguments asfor Theorem 5.11 to get rid of the term containing g ∈ Test(M ) and from above topass from ∇hj to Vj , j ∈ 1, . . . , N, we get

∣∣∣N∑

j=1

Hess f(Vj , Vj) − (∆f)2∣∣∣2

=∣∣∣N∑

j=1

Hess f(Vj , Vj) − 1N

(∆f)2N∑

j=1

|Vj |2∣∣∣2

= 0 m-a.e. on B.

This provides the assertion by the arbitrariness of B.

Corollary 5.15. For every k ∈ N with k > ⌊N⌋,

L2(ΛkT ∗M ) = 0.Proposition 5.14 opens the door for considering an N -Ricci tensor on BE2(κ,N)

spaces with N < ∞, see Subsection 8.2.3 below.

Remark 5.16. It is an interesting task to carry out a detailed study of sufficient andnecessary conditions for the constancy of the local dimension of L2(TM ) as well asits maximality. Natural questions in this respect are the following.

a. Under which hypotheses does there exist d ∈ 1, . . . , ⌊N⌋ such that

m[Ecd

]= 0? (5.18)

b. If (5.18) holds for some d ∈ 1, . . . , ⌊N⌋, which conclusions can be drawnfor the space (M ,E,m)? Does it satisfy BE2(κ, d)?

c. What happens if N is an integer and d = N in (5.18)?In [65], a general class of examples which obey BE2(K,N), K ∈ R and N ∈

[1,∞), but do not have constant local dimension has been pointed out. The latteralready happens, for instance, for metric measure spaces obtained by gluing togethertwo compact pointed Riemannian manifolds at their base points. The point is thatsuch a space does not satisfy the Sobolev-to-Lipschitz property, hence cannot beRCD(K ′,∞) for any K ′ ∈ R [4, Thm. 6.2].

On the other hand, some existing results in the framework of RCD(K,N) spaces(M , d,m), K ∈ R and N ∈ [1,∞), are worth mentioning.

Originating in [82], a. has completely been solved in [22, Thm. 0.1]. (This resultfrom [22] uses optimal transport tools. See [53] for the connections of the latterto [45].) However, it is still unknown what the corresponding value of d really is,e.g. whether it generally coincides with the Hausdorff dimension of (M , d).

60 MATHIAS BRAUN

Questions b. and c. are still subject to high research interest. Results in thisdirection have been initiated in [34]. Indeed, every RCD(K,N) space with integerNand m[Ec

N ] = 0 is weakly noncollapsed [34, Def. 1.10] by [34, Thm. 1.12, Rem. 1.13].In fact, it is conjectured in [34, Rem. 1.11] that every such weakly noncollapsedRCD(K,N) space is noncollapsed [34, Def. 1.1], i.e. m is a constant multiple ofHN . This conjecture is true if M is compact [66, Cor. 1.3]. Lastly, we mention for

completeness that the second question in b. is true for general RCD(K,N) spaces[34, Rem. 1.13].

Since to our knowledge, the result from [22] in Remark 5.16 has not yet beenconsidered in the context of Hino indices, let us phrase it separately according toour compatibility result in Corollary 2.8 to bring it to a broader audience.

Corollary 5.17. Let (M ,E,m) be the Dirichlet space induced by an RCD(K,N)space (M , d,m), K ∈ R and N ∈ [1,∞). Then there exists d ∈ 1, . . . , ⌊N⌋ suchthat the pointwise Hino index of E is m-a.e. constantly equal to d.

5.4. Calculus rules. This section contains calculus rules for the Hessian as well aspreparatory material, such as different function spaces, required to develop them.We shortly comment on the two major challenges in establishing these.

First, the calculus rules in Subsection 5.4.3 below — which easily hold for testfunctions — do not transfer to arbitrary elements in D(Hess) in general, at least notby approximation. In fact, in general Test(M ) is even not dense in D(Hess): on acompact smooth Riemannian manifold M with boundary, any nonconstant, affinef : M → R belongs to D(Hess), but as a possible limit of elements of Test(M )in D(Hess) it would necessarily have s-a.e. vanishing normal derivative at ∂M byLemma 3.15, Example 3.9 and the trace theorem from Proposition 1.2 for k := p :=2 and E := M × R, see also [88, p. 32]. Hence, in Definition 5.27 we consider theclosure Dreg(Hess) of Test(M ) in D(Hess). Many calculus rules will “only” hold forthis class.

Second, to prove a product rule for the Hessian, Proposition 5.30, similarly toDefinition 5.2 above one would try to define the space W 2,1(M ) of all f ∈ L1(M )having a gradient ∇f ∈ L1(TM ) and a Hessian Hess f ∈ L1((T ∗)⊗2M ). How-ever, we refrain from doing so, since in this case the term

⟨∇f,∇〈∇g1,∇g2〉

⟩,

g1, g2 ∈ Test(M ), appearing in the defining property of Hess f in Definition 5.2cannot be guaranteed to have any integrability by the lack of L∞-bounds on∇〈∇g1,∇g2〉. (In view of Proposition 6.11 and Lemma 6.13, this would amountto the strong requirement of bounded Hessians of g1 and g2. Compare with [47,Rem. 4.10, Rem. 4.11, Rem. 4.13].) A similar issue arises for the Hessian chain rulein Proposition 5.31. Since the requirement ∇f ∈ L2(M ) thus cannot be dropped,we are forced to work with the space D2,2,1(Hess) introduced in Definition 5.28.

5.4.1. First order spaces.

Definition 5.18. We define the space G to consist of all f ∈ L1(M ) for whichthere exists η ∈ L1(T ∗M ) such that

ˆ

M

η(∇g)h dm = −ˆ

M

f div(h∇g) dm

for every g ∈ TestL∞(M ) and every h ∈ Test(M ). If such an η exists, it is unique,denoted by d1f and termed the differential of f .

The defining equality makes sense by Lemma 3.15. The uniqueness statementfollows from weak∗ density of Test(TM ) in L∞(TM ) as discussed in Section 4.3.Therefore, d1 becomes a linear operator on G, which turns the latter into a realvector space. Since both sides of the defining property for d1 in Definition 5.18


are strongly L1-continuous in f and η, respectively, for fixed g ∈ TestL∞(M ) andh ∈ Test(M ), G is a Banach space w.r.t. the norm ‖ · ‖G given by

‖f‖G := ‖f‖L1(M) + ‖d1f‖L1(T∗M).

Products of test functions belong to G by Lemma 3.15. In fact, by Lemma 4.17,G ∩ F and G ∩ Test(M ) are dense in F. The subsequent definition — that weintroduce since we do not know if G∩ Test(M ) is dense in G — is thus non-void.

Definition 5.19. We define the space Greg ⊂ G by

Greg := cl‖·‖G

[G∩ Test(M )

].

One checks through Lemma 4.17 and Lemma 4.18 that Greg is dense in L1(M ).A priori, the differential from Definition 5.18 could differ from the differential d

on Fe from Definition 2.6. However, these objects agree on the intersection G∩ Fe.In particular, in what follows we simply write df in place of d1f for f ∈ G althoughthe axiomatic difference should always be kept in mind.

Lemma 5.20. If f ∈ G∩ Fe, then

d1f = df.

Proof. Given any n ∈ N, the function fn := maxminf, n,−n ∈ Fe belongs toL1(M ) ∩ L∞(M ) and thus to F by Proposition 1.8. Let g ∈ TestL∞(M ) and h ∈Test(M ) be arbitrary. Observe that fn → f in L1(M ) and dfn = 1|f |<n df → dfin L2(T ∗M ) as n → ∞. Since h∇g ∈ DTV(div) ∩ D(div) with n(h∇g) = 0 anddiv(h∇g) = 〈∇h,∇g〉 + h∆g ∈ L∞(M ) by Lemma 3.15,

ˆ

M

d1f(∇g)h dm = −ˆ

M

f div(h∇g) dm

= − limn→∞

ˆ

M

fn div(h∇g) dm

= limn→∞

ˆ

|f |<n

df(∇g)h dm

=ˆ

M

df(∇g)h dm.

The statement follows from the weak∗ density of TestL∞(TM ) in L∞(TM ).

Occasionally we adopt the dual perspective of df for a given f ∈ G similarly toDefinition 3.4 — more precisely, under the compatibility granted by Lemma 5.20and keeping in mind Subsection 4.3.2, define ∇f ∈ L1(TM ) by

∇f := (df)♯.

Lemma 5.21. For every f ∈ Greg and every g ∈ F such that dg ∈ L∞(T ∗M ) and∆g ∈ L∞(M ), we have

ˆ

M

f ∆g dm = −ˆ

M

df(∇g) dm.

Proof. Given a sequence (fn)n∈N in Test(M ) ∩ G converging f in G, since fn ∈ F

for every n ∈ N, by Lemma 5.20 we haveˆ

M

f ∆g dm = limn→∞

ˆ

M

fn ∆g dm

= − limn→∞

ˆ

M

dfn(∇g) dm

= −ˆ

M

df(∇g) dm.

62 MATHIAS BRAUN

Remark 5.22. It is unclear to us whether the integration by parts formula fromLemma 5.21 holds if merely f ∈ G. The subtle point is that in general, we are notable to get rid of the zeroth order term h in Definition 5.18 — Lemma 1.9 doesnot give global first order controls. Of course, what still rescues this importantidentity (see e.g. Proposition 6.24 below) for Greg-functions is its validity for theapproximating test functions, and Lemma 5.20.

By the same reason, in our setting we lack an analogue of [45, Prop. 3.3.18],see also [46, Le. 6.2.26], which grants F-regularity of f ∈ G∩ L2(M ) as soon asdf ∈ L2(T ∗M ). Compare with Remark 7.3 and also Remark 8.41 below.

Both statements are clearly true if M is intrinsically complete.

In line with Remark 5.22, the following proposition readily follows from corre-sponding properties of test functions from Proposition 2.11, Lemma 5.20 and anargument as for [45, Thm. 2.2.6].

Proposition 5.23. The following properties hold for the space Greg and the differ-ential d.

(i) Locality. For every f, g ∈ Greg,

1f=g df = 1f=g dg.

(ii) Chain rule. For every f ∈ Greg and every L1-negligible Borel set C ⊂ R,

1f−1(C) df = 0.

In particular, for every ϕ ∈ Lip(R), we have ϕ f ∈ Greg with

d(ϕ f) =[ϕ′ f

]df,

where the derivative ϕ′ f is defined arbitrarily on the intersection of theset of non-differentiability points of ϕ with the image of f .

(iii) Leibniz rule. For every f, g ∈ Greg ∩ L∞(M ), f g ∈ Greg with


Remark 5.24. We do not know if Proposition 5.23 holds for functions merely in G.E.g., unlike the identification result Lemma 5.20 it is unclear whether the gradientestimate from (4.2) holds for every f ∈ G or whether pt maps G into G, t > 0.

Lemma 5.25. If f ∈ Fe ∩ L1(M ) with df ∈ L1(T ∗M ), then f ∈ Greg.

Proof. Define fn ∈ Fe ∩L1(M )∩L∞(M ) by fn := maxminf, n,−n, n ∈ N. ByProposition 2.11, we have dfn = 1|f |<n df ∈ L1(T ∗M ), and therefore dfn → dfin L1(T ∗M ) as n → ∞.

We claim that fn ∈ Greg for every n ∈ N, which then readily yields the conclusionof the lemma. Indeed, given n ∈ N and t > 0, we have ptfn ∈ Test(M ) ∩ L1(M ),and in fact ptfn ∈ G by Lemma 3.15. Since ptfn → fn in Fas t → 0, by Lebesgue’stheorem and the assumption that f ∈ L1(M ) it follows that ptfn → fn in L1(M ) ast → 0. Moreover, since dptfn → dfn in L2(T ∗M ) as t → 0 and since (dptfn)t∈[0,1]

is bounded in L1(T ∗M ) thanks to (4.2) and exponential boundedness of (pκt )t≥0 inL1(M ) [41, Rem. 2.14], applying Lebesgue’s theorem again we obtain that dptfn →dfn in L1(T ∗M ) as t → 0.

Lemma 5.26. For every f, g ∈ F we have f g ∈ Greg with


Proof. Again we define fn, gn ∈ F∩ L∞(M ) by fn := maxminf, n,−n andgn := maxming, n,−n, n ∈ N. By Proposition 2.11 and Remark 2.12, we havefn gn ∈ Fe ∩ L1(M ) with

d(fn gn) = fn dgn + gn dfn = fn 1|g|<n dg + gn 1|f |<n df


whence fn gn ∈ Greg for every n ∈ N by Lemma 5.25. Since fn gn → f g in L1(M )and fn 1|g|<n dg+gn 1|f |<n df → f dg+g df in L1(T ∗M ) as n → ∞, respectively,the conclusion follows.

5.4.2. Second order spaces.

Definition 5.27. We define the space Dreg(Hess) ⊂ D(Hess) as

Dreg(Hess) := cl‖·‖D(Hess)Test(M ).

By Corollary 5.12, we have

Dreg(Hess) = cl‖·‖D(Hess)D(∆).

This space plays an important role in Proposition 5.32, but also in later discussionson calculus rules for the covariant derivative, see Section 6.2.

As indicated in the beginning of Section 5.4, we consider the space D2,2,1(Hess)of F-functions with an L1-Hessian, which seems to be the correct framework forProposition 5.30 and Proposition 5.31 below. (The numbers in the subscript denotethe degree of integrability of f , df and Hess f , respectively, f ∈ D2,2,1(Hess).) Asafter Example 5.1, one argues that the defining property is well-defined.

Definition 5.28. We define the space D2,2,1(Hess) to consist of all f ∈ F for whichthere exists A ∈ L1((T ∗)⊗2M ) such that for every g1, g2, h ∈ Test(M ),

2ˆ

M

hA(∇g1,∇g2) dm

= −ˆ

M


M


−ˆ

M

h⟨∇f,∇〈∇g1,∇g2〉

⟩dm.

In case of existence, A is unique, denoted by Hess1 f and termed the Hessian of f .

By the weak∗ density of Test(T⊗2M ) in L∞(T⊗2M ), the uniqueness statementin Definition 5.28 is indeed true given that such an element A exists. Furthermore,if f ∈ D2,2,1(Hess) ∩ D(Hess) then of course

Hess1 f = Hess f.

We shall thus simply write Hess in place of Hess1 also for functions in D2,2,1(Hess)without further notice. D2,2,1(Hess) is a real vector space, and Hess is a linearoperator on it.

The space D2,2,1(Hess) is endowed with the norm ‖ · ‖D2,2,1(Hess) given by

‖f‖D2,2,1(Hess) := ‖f‖L2(M) + ‖df‖L2(T∗M) + ‖Hess f‖L1((T∗)⊗2M).

Since both sides of the defining property in Definition 5.28 are continuous in f andAw.r.t. convergence in Fand L1((T ∗)⊗2M ), respectively, this norm turns D2,2,1(Hess)into a Banach space. It is also separable, since the map Id×d×Hess : D2,2,1(Hess) →L2(M ) × L2(T ∗M ) × L1((T ∗)⊗2M ) is an isometry onto its image, where the latterspace is endowed with the usual product norm

‖(f, ω,A)‖ := ‖f‖L2(M) + ‖ω‖L2(T∗M) + ‖A‖L1((T∗)⊗2M)

which is separable by Lemma 2.7 and the discussion from Subsection 4.3.2.

64 MATHIAS BRAUN

5.4.3. Calculus rules for the Hessian. After having introduced the relevant spaces,we finally proceed with the calculus rules for functions in D2,2,1(Hess).

The next lemma will be technically useful. For h ∈ Test(M ), the asserted equalityis precisely the defining property of Hess f stated in Definition 5.2. The general caseh ∈ F∩ L∞(M ) follows by replacing h by pth ∈ Test(M ), t > 0, and letting t → 0with the aid of Lemma 3.15.

Lemma 5.29. For every f ∈ D(Hess), every g1, g2 ∈ Test(M ) and every h ∈F∩ L∞(M ),

2ˆ

M


= −ˆ

M


M


−ˆ

M

h⟨∇f,∇〈∇g1,∇g2〉

⟩dm.

Proposition 5.30 (Product rule). If f, g ∈ D(Hess) ∩ L∞(M ), we have f g ∈D2,2,1(Hess) with

Hess(f g) = gHess f + f Hess g + df ⊗ dg + dg ⊗ df.

Proof. Note that f g ∈ F, and that the r.h.s. of the claimed identity for Hess(f g)defines an element in L1((T ∗)⊗2M ). Now, given any g1, g2, h ∈ Test(M ), byProposition 2.11 and Lemma 3.15 it follows that

−⟨∇(f g),∇g1

⟩div(h∇g2)

= −f 〈∇g,∇g1〉 div(h∇g2) − g 〈∇f,∇g1〉 div(h∇g2)⟨

= −〈∇g,∇g1〉 div(f h∇g2) + h 〈∇g,∇g1〉〈∇f,∇g2〉⟨

− 〈∇f,∇g1〉 div(g h∇g2) + h 〈∇f,∇g1〉〈∇g,∇g2〉 m-a.e.,

and analogously

−⟨∇(f g),∇g2

⟩div(h∇g1)

= −〈∇g,∇g2〉 div(f h∇g1) + h 〈∇g,∇g2〉〈∇f,∇g1〉⟨

− 〈∇f,∇g2〉 div(g h∇g1) + h 〈∇f,∇g2〉〈∇g,∇g1〉 m-a.e.,

while finally

− h⟨∇(f g),∇〈∇g1,∇g2〉

⟩

= −h f⟨∇g,∇〈∇g1,∇g2〉

⟩− h g

⟨∇f,∇〈∇g1,∇g2〉

⟩m-a.e.

Adding up these three identities and using Lemma 5.29 for f with g h in place of hand for g with f h in place of h and by (1.8), the claim readily follows.

Proposition 5.31 (Chain rule). Let f ∈ D(Hess), and let ϕ ∈ C1(R) such thatϕ′ ∈ Lipb(R). If m[M ] = ∞, we also assume ϕ(0) = 0. Then ϕ f ∈ D2,2,1(Hess)as well as

Hess(ϕ f) =[ϕ′′ f

]df ⊗ df +

[ϕ′ f

]Hess f,

where ϕ′′f is defined arbitrarily on the intersection of the set of non-differentiabilitypoints of ϕ′.

Proof. By Proposition 2.11 and the boundedness of ϕ′, we have ϕ f ∈ F. Ther.h.s. of the claimed identity defines an object in L1(M ). Now, given any g1, g2, h ∈Test(M ), as in the proof of Proposition 5.30 we have

−⟨∇(ϕ f),∇g1

⟩div(h∇g2)


= −[ϕ′ f

]〈∇f,∇g1〉 div(h∇g2)

= −〈∇f,∇g1〉 div([ϕ′ f

]h∇g2

)

+ h[ϕ′′ f

]〈∇f,∇g1〉〈∇f,∇g2〉 m-a.e.,

and analogously

−⟨∇(ϕ f),∇g2

⟩div(h∇g1)

= −〈∇f,∇g2〉 div([ϕ′ f

]h∇g1

)

+ h[ϕ′′ f

]〈∇f,∇g2〉〈∇f,∇g1〉 m-a.e.,

while finally

−h⟨∇(ϕ f),∇〈∇g1,∇g2〉

⟩= −h

[ϕ′ f

] ⟨∇f,∇〈∇g1,∇g2〉

⟩m-a.e.

Adding up these three identities and using Lemma 5.29 for f with h [ϕ′ f ] in placeof h and by (1.8), the claim readily follows.

Proposition 5.32 (Product rule for gradients). The following properties hold forevery f ∈ D(Hess) and every g ∈ Dreg(Hess).

(i) We have 〈∇f,∇g〉 ∈ G with

d〈∇f,∇g〉 = Hess f(∇g, ·) + Hess g(∇f, ·).(ii) For every g1, g2 ∈ Dreg(Hess),

2 Hess f(∇g1,∇g2) =⟨∇g1,∇〈∇f,∇g2〉

⟩+

⟨∇g2,∇〈∇f,∇g1〉

⟩

−⟨∇f,∇〈∇g1,∇g2〉

⟩m-a.e.

(iii) If f ∈ Dreg(Hess) as well, then 〈∇f,∇g〉 ∈ Greg.

Proof. We first treat item (i) under the additional assumption that g ∈ Test(M ).Let g′ ∈ TestL∞(M ) and h ∈ Test(M ). Let (fn)n∈N be a sequence in Test(M )which converges to f in F. Then by Definition 5.2,

ˆ

M

h[Hess f(∇g,∇g′) + Hess g(∇f,∇g′)

]dm

= limn→∞

ˆ

M

h[Hess f(∇g,∇g′) + Hess g(∇fn,∇g′)

]dm

= −12

ˆ

M

〈∇f,∇g〉 div(h∇g′) dm − 12

ˆ

M

〈∇f,∇g′〉 div(h∇g) dm

− 12

ˆ

M

h⟨∇f,∇〈∇g,∇g′〉

⟩dm

− 12

limn→∞

ˆ

M

〈∇g,∇fn〉 div(h∇g′) dm

− 12

limn→∞

ˆ

M

〈∇g,∇g′〉 div(h∇fn) dm

− 12

limn→∞

ˆ

M

h⟨∇g,∇〈∇fn,∇g′〉

⟩dm

= −ˆ

M

〈∇f,∇g〉 div(h∇g′) dm − 12

ˆ

M

〈∇f,∇g′〉 div(h∇g) dm

+12

limn→∞

ˆ

M

〈∇fn,∇g′〉 div(h∇g) dm

− 12

ˆ

M

h⟨∇f,∇〈∇g,∇g′〉

⟩dm

+12

limn→∞

ˆ

M

h⟨∇〈∇g,∇g′〉,∇fn

⟩dm

66 MATHIAS BRAUN

= −ˆ

M

〈∇f,∇g〉 div(h∇g′) dm.

In the second last step, we used Lemma 3.15 and the fact that 〈∇g,∇g′〉 ∈ F

as well as 〈∇fn,∇g′〉 ∈ F for every n ∈ N provided by Proposition 4.20. SinceHess f(∇g, ·) + Hess g(∇f, ·) ∈ L1(T ∗M ), the claim follows from the definition of Gin Definition 5.18.

To cover the case of general g ∈ Dreg(Hess), simply observe that given any f ∈D(Hess), g′ ∈ TestL∞(M ) and h ∈ Test(M ), both sides of the above computationare continuous in g w.r.t. ‖ · ‖D(Hess).

Point (ii) easily follows by expressing each summand in the r.h.s. of the claimedidentity in terms of the formula from (i), and the symmetry of the Hessian knownfrom Theorem 5.3.

We finally turn to (iii). If f, g ∈ Test(M ), then 〈∇f,∇g〉 ∈ F∩ L1(M ) thanksto Proposition 4.20, while d〈∇f,∇g〉 ∈ L1(T ∗M ) by (i). The Greg-regularity of〈∇f,∇g〉 thus follows from Lemma 5.25. For general f, g ∈ Dreg(Hess), let (fn)n∈N

and (gn)n∈N be two sequences in Test(M ) such that fn → f and gn → g in D(Hess)as n → ∞, respectively. Then clearly 〈∇fn,∇gn〉 → 〈∇f,∇g〉 in L1(M ) as n → ∞,while Hess fn(∇gn, ·) + Hess gn(∇fn, ·) → Hess f(∇g, ·) + Hess g(∇f, ·) in L1(T ∗M )as n → ∞. By the Greg-regularity of 〈∇fn,∇gn〉 for every n ∈ N already shownabove, the proof is terminated.

Combining the last two items of Proposition 5.32 with Proposition 5.23 thusentails the following locality property.

Lemma 5.33 (Locality of the Hessian). For every f, g ∈ Dreg(Hess),

1f=g Hess f = 1f=g Hess g.

6. Covariant derivative

6.1. The Sobolev space W 1,2(TM ). In a similar kind as in Example 5.1, thesmooth context motivates how we should define a nonsmooth covariant derivativeacting on vector fields, having now the notion of Hessian at our disposal.

Example 6.1. Let M be a Riemannian manifold with boundary. The covariantderivative ∇X of X ∈ Γ(TM ) is uniquely defined by the pointwise relation

⟨∇∇g1X,∇g2

⟩= ∇X : (∇g1 ⊗ ∇g2)

=⟨∇〈X,∇g2〉,∇g1

⟩− Hess g2(∇g1, X)

(6.1)

for every g1, g2 ∈ C∞c (M ), see e.g. [46, Thm. 6.2.1]. That is, ∇ is tensorial in its

first and derivative in its second component, torsion-free and metrically compatibleif and only if the second equality in (6.1) holds for every g1 and g2 as above. Thisyields an alternative definition of the Levi-Civita connection ∇ in place of Koszul’sformula, see e.g. [84, p. 25], which does not use Lie brackets. (Indeed, in our settingit is not even clear how to define the Lie bracket without covariant derivatives.)

As in Example 5.1, ∇ is still uniquely determined on Γ(TM ) by the validity of(6.1) for every g1, g2 ∈ C∞

c (TM ) for which

〈∇g1, n〉 = 〈∇g2, n〉 = 0 on ∂M .

For such g1 and g2, we integrate (6.1) against hm for a given h ∈ C∞c (M ) to obtain,

using again the regularity discussion around (5.3),ˆ

M

h∇X : (∇g1 ⊗ ∇g2) dv

= −ˆ

M

〈X,∇g2〉 div(h∇g1) dv −ˆ

M

hHess g2(X,∇g1) dv.(6.2)


This identity characterizes ∇X by the existence of a smooth extension to ∂M .

In our setting, the r.h.s. of (6.2) — with v replaced by m — still makes sense forg1, g2, h ∈ Test(M ) and is compatible with the smooth case (in the sense that noboundary terms show up in the nonsmooth terminology of Definition 3.8), whichis argued as in the paragraph after Example 5.1. Indeed, for X ∈ L2(TM ) wehave 〈X,∇g2〉 ∈ L2(M ) as well as div(h∇g1) = 〈∇h,∇g1〉 + h∆g1 ∈ L2(M ) byLemma 3.15, while the second integral on the r.h.s. of (6.2) is trivially well-defined.

This motivates the subsequent definition.

Definition 6.2. The space W 1,2(TM ) is defined to consist of all X ∈ L2(TM ) forwhich there exists T ∈ L2(T⊗2M ) such that for every g1, g2, h ∈ Test(M ),

ˆ

M

hT : (∇g1 ⊗ ∇g2) dm

= −ˆ

M

〈X,∇g2〉 div(h∇g1) dm −ˆ

M

hHess g2(X,∇g1) dm.

In case of existence, the element T is unique, denoted by ∇X and termed the co-variant derivative of X.

Arguing as for the Hessian after Definition 5.2, the uniqueness statement inDefinition 6.2 is derived. In particular, W 1,2(TM ) constitutes a vector space andthe covariant derivative ∇ is a linear operator on it. Further properties can beconsulted in Theorem 6.3 below.

The space W 1,2(TM ) is endowed with the norm ‖ · ‖W 1,2(TM) given by∥∥X

∥∥2

W 1,2(TM):=

∥∥X∥∥2

L2(TM)+

∥∥∇X∥∥2

L2(T⊗2M).

We also define the covariant functional Ecov : L2(TM ) → [0,∞] by

Ecov(X) :=

ˆ

M

∣∣∇X∣∣2

HSdm if X ∈ W 1,2(TM ),

∞ otherwise.(6.3)

Theorem 6.3. The space W 1,2(TM ), the covariant derivative ∇ and the functionalEcov possess the following properties.

(i) W 1,2(TM ) is a separable Hilbert space w.r.t. ‖ · ‖W 1,2(TM).(ii) The covariant derivative ∇ is a closed operator. That is, the image of the

map Id × ∇ : W 1,2(TM ) → L2(TM ) × L2(T⊗2M ) is a closed subspace ofL2(TM ) × L2(T⊗2M ).

(iii) For every f ∈ Test(M ) and every g ∈ Test(M ) ∪ R 1M , we have g∇f ∈W 1,2(TM ) with

∇(g∇f) = ∇g ⊗ ∇f + g (Hess f)♯,

with the interpretation ∇1M := 0. In particular Reg(TM ) ⊂ W 1,2(TM ),and W 1,2(TM ) is dense in L2(TM ).

(iv) The functional Ecov is lower semicontinuous, and for every X ∈ L2(TM )we have the duality formula

Ecov(X) = sup

−2n∑

i=1

ˆ

M

〈X,Zi〉 div Yi dm − 2n∑

i=1

ˆ

M

∇Zi : (Yi ⊗X) dm

−ˆ

M

∣∣∣n∑

i=1

Yi ⊗ Zi

∣∣∣2

dm : n ∈ N, Yi, Zi ∈ Test(TM ).

68 MATHIAS BRAUN

Proof. Item (ii) is addressed by observing that given any g1, g2, h ∈ Test(M ), bothsides of the defining property of ∇ in Definition 6.2 are continuous in X and Tw.r.t. weak convergence in L2(TM ) and L2(T⊗2M ), respectively.

For (i), it is clear from (ii) and the trivial parallelogram identity of ‖ · ‖W 1,2(TM)

that W 1,2(TM ) is a Hilbert space. To prove its separability, we endow L2(TM ) ×L2(T⊗2M ) with the product norm ‖ · ‖ given by

‖(X,T )‖2 :=∥∥X

∥∥2

L2(TM)+

∥∥T∥∥2

L2(T⊗2M).

Recall from Lemma 2.7, Proposition 1.24 and the discussions in Subsection 1.4.5and Section 4.3 that L2(TM ) and L2(T⊗2M ) are separable Hilbert spaces, and sois their Cartesian product. As Id × ∇ : W 1,2(TM ) → L2(TM ) × L2(T⊗2M ) is abijective isometry onto its image, the proof of (i) is completed.

Item (iii) for g := 1M follows from Proposition 5.32. Indeed, given any g1, g2, h ∈Test(M ), by definition of G we have

ˆ

M


=ˆ

M

h⟨∇〈∇f,∇g2〉,∇g1

⟩dm −

ˆ

M

hHess g2(∇f,∇g2) dm

= −ˆ

M


M

hHess g2(∇f,∇g2) dm.

The argument for g ∈ Test(M ) follows similar lines. To prove it, let g1, g2, h ∈Test(M ). Recall from Proposition 4.20 that 〈∇f,∇g2〉 ∈ F. Since additionally〈∇f,∇g2〉 ∈ L∞(M ), we also have g 〈∇f,∇g2〉 ∈ F by the Leibniz rule for thegradient — compare with Proposition 2.11 — which, also taking into accountProposition 5.32 and Lemma 5.20, yields

d[g 〈∇f,∇g2〉

]= 〈∇f,∇g2〉 dg + gHess f(∇g2, ·) + gHess g2(∇f, ·).

This entails that

−ˆ

M

g 〈∇f,∇g2〉 div(h∇g1) dm −ˆ

M

h gHess g2(∇f,∇g1) dm

=ˆ

M

d[g 〈∇f,∇g2〉

](h∇g1) dm −

ˆ

M

h gHess g2(∇f,∇g1) dm

=ˆ

M

h[〈∇g,∇g1〉〈∇f,∇g2〉 + gHess f(∇g1,∇g2)

]dm,

which is the claim by the definition (1.8) of the pointwise tensor product.Concerning (iv), the lower semicontinuity of Ecov follows from the fact that

bounded sets in the Hilbert space L2(T⊗2M ) are weakly relatively compact, andfrom the closedness of ∇ that has been shown in (ii).

The proof of the duality formula for Ecov is analogous to the one for the dualityformula for E2 in Theorem 5.3. Details are left to the reader.

Remark 6.4. Similarly to Remark 5.6 and motivated by the RCD∗(K,N) result [54,Prop. 3.12], K ∈ R and N ∈ [1,∞), in practice the notion of covariant derivativefrom Definition 6.2 should only depend on conformal transformations of 〈·, ·〉, butbe independent of drift transformations of m.

6.2. Calculus rules. In this section, we proceed in showing less elementary — stillexpected — calculus rules for the covariant derivative. Among these, we especiallyregard Proposition 6.11 and Lemma 6.13 as keys for the functionality of our secondorder axiomatization which also potentially involves “boundary contributions”.


6.2.1. Some auxiliary spaces of vector fields. As in Section 5.4, we introduce twoSobolev spaces of vector fields we use in the sequel. In fact, the space H1,2(TM )which is introduced next plays a dominant role later, see e.g. Section 6.3.

Definition 6.5. We define the space H1,2(TM ) ⊂ W 1,2(TM ) as

H1,2(TM ) := cl‖·‖W 1,2(T M)Reg(TM ).

By Theorem 6.3, we see that ∇Dreg(Hess) ⊂ H1,2(TM ).H1,2(TM ) is in general a strict subset of W 1,2(TM ). For instance, on compact

Riemannian manifolds with boundary, this follows as in the beginning of Section 5.4by (3.2), Lemma 3.15, Example 3.9 and Proposition 1.2 for E := TM .

Remark 6.6. For noncollapsed mGH-limits of Riemannian manifolds without bound-ary under uniform Ricci and diameter bounds, it is proved in [64, Prop. 4.5] thatH1,2(TM ) = W 1,2(TM ). This does not conflict with our above argument involvingthe presence of a boundary, since spaces “with boundary” are known not to appearas noncollapsed Ricci limits [27, Thm. 6.1].

Remark 6.7. Besides technical reasons, Definition 6.5 has the advantage that itincludes both gradient vector fields of test functions as well as general elements ofTest(TM ) in the calculus rules below. See also Chapter 8.

The “H1,2-space” of vector fields introduced in [45, Def. 3.4.3] is rather givenby cl‖·‖W 1,2(T M)

Test(TM ), which — in our generality — is a priori smaller thanH1,2(TM ). The converse inclusion seems more subtle: as similarly encountered inRemark 5.22, the issue is that we do not really know how to pass from zeroth-orderfactors belonging to Test(M ) to constant ones in a topology which also takes intoaccount “derivatives” of vector fields in our generality. However, since “⊃” holdse.g. if M is intrinsically complete by Theorem 6.3, Definition 6.5 and the approachfrom [45, Def. 3.4.3] give rise to the same space on RCD(K,∞) spaces, K ∈ R.

As in Definition 5.28, in view of Lemma 6.9 we introduce a space intermediatebetween W 1,2(TM ) and “W 1,1(TM )”, a suitable space (that we do not define) ofall X ∈ L1(TM ) with covariant derivative ∇X in L1(TM ). The problem arisingin defining the latter, compare with Definition 6.2, is that we do not know if thereexists a large enough class of g2 ∈ Test(M ) such that Hess g2 ∈ L∞((T ∗)⊗2M ).

Definition 6.8. The space W (2,1)(TM ) is defined to consist of all X ∈ L2(TM )for which there exists T ∈ L1(T⊗2M ) such that for every g1, g2, h ∈ Test(M ),

ˆ

M

hT : (∇g1 ⊗ ∇g2) dm

= −ˆ

M

〈X,∇g2〉 div(h∇g1) dm −ˆ

M

hHess g2(X,∇g1) dm.

In case of existence, the element T is unique, denoted by ∇1X and called the co-variant derivative of X.

Indeed, the uniqueness follows from weak∗ density of Test(T⊗2M ) in L∞(T⊗2M ).In particular, W (2,1)(TM ) is clearly a vector space, and ∇1 is a linear operator onit. By definition, for X ∈ W (2,1)(TM ) ∩W 1,2(TM ) we furthermore have

∇1X = ∇X,whence we subsequently write ∇ in place of ∇1 as long as confusion is excluded.

We endow W (2,1)(TM ) with the norm ‖ · ‖W (2,1)(TM) given by

‖X‖W (2,1)(TM) := ‖X‖L2(TM) + ‖∇X‖L1(T⊗2M).

Since both sides of the defining property in Definition 6.8 are continuous w.r.t. weakconvergence in X and T , respectively, this norm turns W (2,1)(TM ) into a Banach

70 MATHIAS BRAUN

space. Moreover, since the map Id × ∇ : W (2,1)(TM ) → L2(TM ) × L1(T⊗2M ) is abijective isometry onto its image — where L2(TM ) × L1(T⊗2M ) is endowed withthe usual product norm — Lemma 2.7, Proposition 1.24 and the discussion fromSubsection 4.3.4 show that W (2,1)(TM ) is separable.

6.2.2. Calculus rules for the covariant derivative.

Lemma 6.9 (Leibniz rule). Let X ∈ W 1,2(TM ) and f ∈ F∩ L∞(M ). Thenf X ∈ W (2,1)(TM ) and

∇(f X) = ∇f ⊗X + f ∇X.

Proof. We first assume that f ∈ Test(M ). Given any g1, g2, h ∈ Test(M ), we haveh f ∈ Test(M ) since Test(M ) is an algebra. By Lemma 3.15 and the Leibniz rulefor the gradient, Proposition 2.11, we obtain

div(h f ∇g1) = h 〈∇f,∇g1〉 + f div(h∇g1) m-a.e.

Hence, by definition of ∇X from Definition 6.2 and Theorem 5.11,ˆ

M

h (f ∇X) : (∇g1 ⊗ ∇g2) dm

= −ˆ

M

〈X,∇g2〉 div(h f ∇g1) dm −ˆ

M

h f Hess g2(X,∇g1) dm

= −ˆ

M

h 〈∇f,∇g1〉〈X,∇g2〉 dm −ˆ

M

〈f X,∇g2〉 div(h∇g1) dm

−ˆ

M

hHess g2(f X,∇g1) dm.

Rearranging terms according to Definition 6.8 yields the claim.For general f ∈ F∩ L∞(M ), note that on the one hand ptf ∈ Test(M ) and

‖ptf‖L∞(M) ≤ ‖f‖L∞(M) holds for every t > 0, and on the other hand ptf → f

in F as t → 0. Thus, we easily see that ptf X → f X in L2(TM ) as well as∇(ptf X) = ∇ptf ⊗ X + ptf ∇X → f ⊗ X + f ∇X in L1(T⊗2M ) as t → 0, andthis suffices to conclude the proof.

Remark 6.10. Elementary approximation arguments, also using Lemma 4.18 for (ii),entail the following two H1,2-variants of Lemma 6.9.

(i) If f ∈ Test(M ) and X ∈ H1,2(TM ), then f X ∈ H1,2(TM ) and

∇(f X) = ∇f ⊗X + f ∇X.(ii) The same conclusion as in (i) holds if merely f ∈ F∩ L∞(M ) and X ∈

H1,2(TM ) ∩ L∞(TM ).

In view of the important metric compatibility of ∇ discussed in Proposition 6.11as well as its consequences below, we shall first make sense of directional derivativesof a given X ∈ W 1,2(TM ) in the direction of Z ∈ L0(TM ). There exists a uniquevector field ∇ZX ∈ L0(TM ) which satisfies

〈∇ZX,Y 〉 = ∇X : (Z ⊗ Y ) m-a.e. (6.4)

for every Y ∈ L0(TM ). Indeed, the r.h.s. is a priori well-defined for every Y, Z ∈L0(TM ) for which Z⊗Y ∈ L2(T⊗2M ), but this definition can and will be uniquelyextended by continuity to a bilinear map ∇X : (·⊗ ·) : L0(TM )2 → L0(M ), and theappropriate existence of ∇ZX is then due to Proposition 1.25. In particular,

|∇ZX | ≤ |∇X | |Z| m-a.e. (6.5)


Proposition 6.11 (Metric compatibility). Let X ∈ W 1,2(TM ) and Y ∈ H1,2(TM ).Then 〈X,Y 〉 ∈ G, and for every Z ∈ L0(TM ),

d〈X,Y 〉(Z) =⟨∇ZX,Y

⟩+

⟨X,∇ZY

⟩m-a.e.

Moreover, if X ∈ H1,2(TM ) as well, then 〈X,Y 〉 ∈ Greg.

Proof. We first prove the claim for Y = ∇f for some f ∈ Test(M ). In this case,given any g, h ∈ Test(M ), by definition of ∇X ,

ˆ

M

h∇X : (∇g ⊗ ∇f) dm

= −ˆ

M

〈X,∇f〉 div(h∇g) dm −ˆ

M

hHess f(X,∇g) dm.

Rearranging terms and recalling Definition 5.18 as well as Theorem 6.3, we see that〈X,∇f〉 ∈ L1(M ) belongs to G and

d〈X,∇f〉 = ∇X : (· ⊗ f) + (Hess f)♯ : (· ⊗ ∇f).

It follows from (6.4) that for every Z ∈ Test(TM ),

d〈X,∇f〉(Z) =⟨∇ZX,∇f

⟩+

⟨∇Z∇f,X

⟩m-a.e.

By L∞-linear extension and the density of Test(TM ) in L2(TM ), this identity holdsin fact for every Z ∈ L2(TM ), and therefore extends to arbitrary Z ∈ L0(TM ) byconstruction of the latter space in Subsection 1.4.3.

The case Y := g∇f , f, g ∈ Test(M ), follows by the trivial m-a.e. identity〈X,Y 〉 = 〈g X,∇f〉, the previously derived identity and Lemma 6.9.

By linearity of the covariant derivative and Theorem 6.3, the foregoing discussionthus readily covers the case of general Y ∈ Reg(TM ).

Now, given any Y ∈ H1,2(TM ), a sequence (Yn)n∈N in Reg(TM ) such that Yn →Y in W 1,2(TM ) as n → ∞ and Z ∈ L2(TM )∩L∞(TM ), we have 〈X,Yn〉 → 〈X,Y 〉and 〈∇ZX,Yn〉+〈∇ZYn, X〉 → 〈∇ZX,Y 〉+〈∇ZY,X〉 in L1(M ) as n → ∞. Passingto the limit in the definition of G, it straightforwardly follows that 〈X,Y 〉 ∈ G. Theclaimed identity for d〈X,Y 〉(Z) follows after passing to suitable subsequences, andit extends to arbitrary Z ∈ L0(TM ) as in the first step of the current proof.

To prove the last claim, suppose first that X,Y ∈ Reg(M ), in which case〈X,Y 〉 ∈ G∩ F by Theorem 6.3, what we already proved, Proposition 4.20 andProposition 2.11. By the duality between L1(TM ) and L∞(T ∗M ) as L∞-modules,see Subsection 4.3.2, and Remark 6.12 below, we deduce that d〈X,Y 〉 ∈ L1(TM ),whence 〈X,Y 〉 ∈ Greg thanks to Lemma 5.20 and Lemma 5.25.

Lastly, given arbitrary X,Y ∈ H1,2(TM ), let (Xn)n∈N and (Yn)n∈N be se-quences in Reg(TM ) that converge to X and Y in W 1,2(TM ), respectively. Thenclearly 〈Xn, Yn〉 → 〈X,Y 〉 in L1(M ) as n → ∞, while Remark 6.12 below ensuresthat d〈Xn, Yn〉 → d〈X,Y 〉 in L1(T ∗M ) as n → ∞, which is the claim.

Remark 6.12. By duality and (6.5), it follows in particular from Proposition 6.11that for every X ∈ W 1,2(TM ) and every Y ∈ H1,2(TM ),

|d〈X,Y 〉| ≤ |∇X |HS |Y | + |∇Y |HS |X | m-a.e.

The following lemma is a version of what is known as Kato’s inequality (for theBochner Laplacian) in the smooth case [58, Ch. 2]. See also [32, Lem. 3.5].

Lemma 6.13 (Kato’s inequality). For every X ∈ H1,2(TM ), |X | ∈ F and∣∣∇|X |

∣∣ ≤∣∣∇X

∣∣HS

m-a.e.

In particular, if X ∈ H1,2(TM ) ∩ L∞(TM ) then |X |2 ∈ F.

72 MATHIAS BRAUN

Proof. We initially prove the frist claim for X ∈ Reg(TM ). Define ϕn ∈ Lip([0,∞))by ϕn(t) := (t+ 1/n2)1/2 − 1/n, n ∈ N. By polarization, Proposition 4.20 and theLeibniz rule stated in Proposition 2.11, we have |X |2 ∈ F, and thus ϕn |X |2 ∈ F

for every n ∈ N. Moreover, ϕn |X |2 → |X | pointwise m-a.e. and in L2(M ) asn → ∞, as well as ϕ′

n(t)2 t ≤ 1/4 for every t ≥ 0 and every n ∈ N. Hence byRemark 6.12 and — since also |X |2 ∈ Greg by Proposition 6.11 — Lemma 5.20,

∣∣∇(ϕn |X |2)∣∣2

=∣∣ϕ′n |X |2

∣∣2 ∣∣∇|X |2∣∣2

≤ 4∣∣ϕ′n |X |2

∣∣2 |X |2∣∣∇X

∣∣2

HS

≤∣∣∇X

∣∣2

HSm-a.e.

(6.6)

Therefore |X | ∈ F by Proposition 1.8, and (ϕn |X |2)n∈N converges F-weakly to|X |. By Mazur’s lemma, suitable convex combinations of elements of (ϕn|X |2)n∈N

converge F-strongly to |X |. The convexity of the carré du champ, see e.g. [6, p. 249],and (6.6) imply the claimed m-a.e. upper bound on

∣∣∇|X |∣∣.

For general X ∈ H1,2(TM ), let (Xn)n∈N be a sequence in Reg(TM ) such thatXn → X in H1,2(TM ) and |Xn| → |X | both pointwise m-a.e. and in L2(M ) asn → ∞. By what we proved above, (|Xn|)n∈N is bounded in F, whence |X | ∈ F

again by Proposition 1.8. Still by what we already proved, every L2-weak limit ofsubsequences of (

∣∣∇|Xn|∣∣)n∈N is clearly no larger than |∇X |HS m-a.e., whence we

obtain∣∣∇|X |

∣∣ ≤ |∇X |HS m-a.e. by Mazur’s lemma. (See also (2.10) in [6].)If X ∈ H1,2(TM ) ∩ L∞(TM ), the F-regularity of |X |2 follows from the one of

|X | and the chain rule in Proposition 2.11.

This Lemma 6.13 has numerous important consequences. First, the F-regularityasserted therein makes it possible in Chapter 8 to pair the function |X | ∈ F, X ∈H1,2(TM ), with the given distribution κ ∈ F

−1qloc(M ). Second, it yields the im-

proved semigroup comparison for the covariant heat flow in Theorem 6.26. Third,it cancels out the covariant term appearing in the definition of the Ricci curvature,Theorem 8.9, leading to a vector q-Bochner inequality for q ∈ [1, 2], Theorem 8.28.Under additional assumptions, the latter again implies improved semigroup com-parison results in Theorem 8.40, this time for the contravariant heat flow. Fourth,it is regarded as the key technical tool in showing that every X ∈ H1,2(TM ) hasa “quasi-continuous representative” similar to [32, Thm. 3.14]. This latter topic isnot addressed here, but the arguments of [32] do not seem hard to adapt to oursetting.

Corollary 6.14. The real-valued function X 7→ 〈κ∣∣ |X |2

⟩defined on H1,2(TM ) is

H1,2-continuous. More precisely, let ρ′ ∈ (0, 1) and α′ ∈ R satisfy Lemma 4.5 forevery µ ∈ κ+, κ−, |κ|. Then for every X,Y ∈ H1,2(TM ),

∣∣⟨µ∣∣ |X |2

⟩1/2 −⟨µ

∣∣ |Y |2⟩1/2∣∣2 ≤ ρ′Ecov(X − Y ) + α′

∥∥X − Y∥∥2

L2(TM).

Proof. Since κ = κ+ − κ−, it suffices to prove the last statement. Given X,Y ∈H1,2(TM ), by Lemma 6.13 we have |X |, |Y |, |X − Y | ∈ F. In particular, the ex-pressions

⟨µ

∣∣ |X |2⟩

and⟨µ

∣∣ |Y |2⟩

make sense. Since∣∣⟨µ

∣∣ |X |2⟩1/2 −

⟨µ

∣∣ |Y |2⟩1/2∣∣2∥∥2

L2

≤⟨µ

∣∣ ∣∣|X | − |Y |∣∣2⟩∥∥2

L2

≤⟨µ

∣∣ |X − Y |2⟩∥∥2

L2

≤ ρ′E(|X − Y |

)+ α′

∥∥X − Y∥∥2

L2(TM)

≤ ρ′Ecov(X − Y ) + α′∥∥X − Y

∥∥2

L2(TM)


by Lemma 4.5 and again thanks to Lemma 6.13, the claim readily follows.

Remark 6.15. We should not expect the function in Corollary 6.14 to admit acontinuous extension to L2(TM ) in general. The reason is once again the case ofcompact Riemannian manifolds M with boundary and, say, with nonnegative Riccicurvature. In this case, κ := ls ∈ K0(M ) by [41, Lem. 2.33, Thm. 4.4], wherel: ∂M → R designates the lowest eigenvalue function of the second fundamentalform I at ∂M . The pairing

⟨κ

∣∣ |X |2⟩

then simply does not make sense for generalvector fields in L2(TM ), which are only defined up to v-negligible sets (unlike theH1,2-case, where the well-definedness of

⟨κ

∣∣ |X |2⟩

comes from the trace theorem).

Lemma 6.16 (Triviality of the torsion tensor). Suppose that f ∈ Dreg(Hess) andX,Y ∈ W 1,2(TM ). Then 〈X,∇f〉, 〈Y,∇f〉 ∈ G and

⟨X,∇〈Y,∇f〉

⟩−

⟨Y,∇〈X,∇f〉

⟩= df(∇XY − ∇YX) m-a.e.

Proof. Proposition 6.11 shows that 〈Y,∇f〉, 〈X,∇f〉 ∈ G with⟨X,∇〈Y,∇f〉

⟩= ∇Y : (X ⊗ ∇f) + Hess f(X,Y )

= df(∇XY ) + Hess f(X,Y ) m-a.e.,⟨Y,∇〈X,∇f〉

⟩= ∇X : (Y ⊗ ∇f) + Hess f(Y,X)

= df(∇YX) + Hess f(X,Y ) m-a.e.

In the last step, we used the symmetry of Hess f from Theorem 5.3. Subtractingthe two previous identities gives the assertion.

Remark 6.17. In the setting of Lemma 6.16, a more familiar way — compared toclassical Riemannian geometry — of writing the stated identity is

X(Y f) − Y (X f) = df(∇XY − ∇YX) m-a.e.

when defining Xf := 〈X,∇f〉, Y f := 〈Y,∇f〉, and accordinglyX(Y f) and Y (X f).

Since dF generates L2(T ∗M ) by Lemma 2.7, it follows by Lemma 4.18 thatdDreg(Hess) generates L2(T ∗M ) as well, in the sense of L∞-modules. Hence, byLemma 6.16, given X,Y ∈ W 1,2(TM ), ∇XY − ∇YX is the unique vector fieldZ ∈ L1(TM ) such that

X(Y f) − Y (X f) = df(Z) m-a.e.

for every f ∈ Dreg(Hess). This motivates the following definition.

Definition 6.18. The Lie bracket [X,Y ] ∈ L1(TM ) of two given vector fieldsX,Y ∈ W 1,2(TM ) is defined by

[X,Y ] := ∇XY − ∇YX.

We terminate with the following locality property. It directly follows from thesecond part of Proposition 6.11 as well as Proposition 5.23.

Lemma 6.19 (Locality of the covariant derivative). If X,Y ∈ H1,2(TM ), then

1X=Y ∇X = 1X=Y ∇Y.

6.3. Heat flow on vector fields. Now we introduce and study the canonical heatflow (tt)t≥0 on L2-vector fields. First, after defining its generator in Definition 6.20,following well-known lines [19] we collect elementary properties of the flow (tt)t≥0 inTheorem 6.23. Then we prove the important semigroup comparison Theorem 6.26between (tt)t≥0 and (pt)t≥0, using Lemma 6.13.

74 MATHIAS BRAUN

6.3.1. Bochner Laplacian. In fact, we have a preliminary choice to make, i.e. eitherto define the Bochner Laplacian on W 1,2(TM ) or on the strictly smaller spaceH1,2(TM ). We choose the latter one since the calculus rules from Section 6.2 aremore powerful, in particular in view of Proposition 6.24 below. Also, no ambiguityoccurs for the background boundary conditions, see Example 6.22.

Definition 6.20. We define D() to consist of all X ∈ H1,2(TM ) for which thereexists Z ∈ L2(TM ) such that for every Y ∈ H1,2(TM ),

ˆ

M

〈Y, Z〉 dm = −ˆ

M

∇Y : ∇X dm.

In case of existence, Z is uniquely determined, denoted by X and termed theBochner Laplacian (or connection Laplacian or horizontal Laplacian) of X.

Observe that D() is a vector space, and that : D() → L2(TM ) is a linearoperator. Both is easy to see from the linearity of the covariant derivative.

We modify the functional from (6.3) with domain W 1,2(TM ) by introducing the“augmented” covariant energy functional Ecov : L2(TM ) → [0,∞] with

Ecov(X) :=

ˆ

M

∣∣∇X∣∣2

HSdm if X ∈ H1,2(TM ),

∞ otherwise.

Clearly, its (non-relabeled) polarization Ecov : H1,2(TM )2 → R is a closed, symmet-ric form, and is the nonpositive, self-adjoint generator uniquely associated to itaccording to [43, Thm. 1.3.1].

A first elementary consequence of this discussion, Lemma 6.13 and Rayleigh’stheorem is the following inequality between the spectral bottoms of ∆ and .

Corollary 6.21. We have

inf σ(−∆) ≤ inf σ(−).

Example 6.22. Let M be a Riemannian manifold with boundary. Recall that everyelement of H1,2(TM ) has s-a.e. vanishing normal component at ∂M by the localversion of Proposition 1.2. In particular, coincides with the self-adjoint realiza-tion in L2(TM ) of the restriction of the usual (non-relabeled) Bochner Laplacian

to the class of compactly supported elements X ∈ Γ(TM ) satisfying the followingmixed boundary conditions on ∂M , see (1.2):

X⊥ = 0,

(∇nX)‖ = 0.(6.7)

Indeed, for any compactly supported X,Y ∈ Γ(TM ),ˆ

M

〈X,Y 〉 dv −ˆ

M

〈X,Y 〉 dv

=ˆ

∂M

〈∇nX,Y 〉 ds −ˆ

∂M

〈X,∇nY 〉 ds

according to the computations carried out in [26, Ch. 2]. The last two integralsvanish under (6.7). Moreover, as remarked in [26] this suffices to recover the definingintegration by parts formula from Definition 6.20.

Let us remark for completeness that in [26], the boundary conditions that aredual to (6.7) have been considered. See also [88, Prop. 1.2.6].


6.3.2. Heat flow and its elementary properties. Analogously to Subsection 1.3.7,we may and will define the heat flow on vector fields as the semigroup (tt)t≥0 ofbounded, linear and self-adjoint operators on L2(TM ) by

tt := et.

Following e.g. [19] or [45, Subsec. 3.4.4], the subsequent elementary propertiesof (tt)t≥0 are readily established.

Theorem 6.23. The following properties of (tt)t≥0 hold for every X ∈ L2(TM )and every t > 0.

(i) The curve t 7→ ttX belongs to C1((0,∞); L2(TM )) with

ddt

ttX = ttX.

(ii) If X ∈ D(), we have

ddt

ttX = ttX.

In particular, we have the identity

tt = tt on D().

(iii) For every s ∈ [0, t],

‖ttX‖L2(TM) ≤ ‖tsX‖L2(TM).

(iv) The function t 7→ Ecov(ttX) belongs to C1((0,∞)), is nonincreasing, andits derivative satisfies

ddtEcov(ttX) = −2

ˆ

M

∣∣ttX∣∣2

dm.

(v) If X ∈ H1,2(TM ), the map t 7→ ttX is contiuous on [0,∞) w.r.t. strongconvergence in H1,2(TM ).

(vi) We have

Ecov(ttX) ≤ 12t

∥∥X∥∥2

L2(TM),

∥∥ttX∥∥2

L2(TM)≤ 1

2t2∥∥X

∥∥2

L2(TM).

6.3.3. Functional inequalities and Lp-properties. The calculus rules from Section 6.2allow us to derive useful functional inequalities of (tt)t≥0 w.r.t. (pt)t≥0. The main re-sult, essentially coming from Proposition 6.11 and Lemma 6.13, is the L1-estimatefrom Theorem 6.26. Lp-consequences of it, p ∈ [1,∞], are stated in Corollary 6.29.

In fact, the latter requires the following L2-version of it in advance for technicalreasons, see Remark 6.27 — in particular, Proposition 6.24 does not follow fromTheorem 6.26 just by Jensen’s inequality for (pt)t≥0.

Proposition 6.24. For every X ∈ L2(TM ) and every t ≥ 0,

|ttX |2 ≤ pt(|X |2

)m-a.e.

Proof. We only prove the nontrivial part in which t > 0. Let φ ∈ TestL∞(M ) benonnegative. Define the function F : [0, t] → R by

F (s) :=ˆ

M

φ pt−s(|tsX |2

)dm =

ˆ

M

pt−sφ |tsX |2 dm.

Of course, F is well-defined. Moreover, s 7→ pt−sφ is Lipschitz continuous as amap from [0, t) into L∞(M ,m) since its derivative at a given s ∈ [0, t) is equalto −∆pt−sφ = −pt−s∆φ. Furthermore, since s 7→ tsX is continuous as a mapfrom [0, t] into L2(TM ) and locally absolutely continuous as a map from (0, t] into

76 MATHIAS BRAUN

L2(TM ), the L1-valued map s 7→ |tsX |2 is continuous on [0, t] and locally absolutelycontinuous on (0, t]. All in all, this discussion implies that F is continuous on[0, t] and locally absolutely continuous on (0, t). By exchanging differentiation andintegration, for L

1-a.e. s ∈ (0, t) we thus get

F ′(s) = −ˆ

M

∆pt−sφ |tsX |2 dm + 2ˆ

M

pt−sf 〈tsX,tsX〉 dm.

Observe that pt−sφ ∈ Test(M ) with ∆pt−sφ = pt−s∆φ ∈ L∞(M ) as wellas |tsX |2 ∈ Greg for every s ∈ (0, t) by Proposition 6.11. By Lemma 5.21 andRemark 6.10, for L

1-a.e. s ∈ (0, t) we have

F ′(s) =ˆ

M

⟨∇pt−sφ,∇|tsX |2

⟩dm − 2

ˆ

M

⟨∇(pt−sφ tsX),∇tsX

⟩dm (6.8)

=ˆ

M


⟩dm − 2

ˆ

M

[∇pt−sφ⊗ tsX ] : ∇tsX dm

− 2ˆ

M

pt−sφ∣∣∇tsX

∣∣2

HSdm

≤ˆ

M


⟩dm − 2

ˆ

M

[∇pt−sφ⊗ tsX ] : ∇tsX dm = 0.

In the last equality, we used Proposition 6.11. Therefore,ˆ

M

φ |ttX |2 dm = F (t) ≤ F (0) =ˆ

M

φ pt(|X |2

)dm.

This proves the claim thanks to the arbitrariness of φ by Lemma 4.18.

In applications, the following corollary of Proposition 6.24 could be useful.

Corollary 6.25. For every X ∈ D(), there exists a sequence (Xn)n∈N in D()∩L∞(TM ) which converges to X in H1,2(TM ) such that in addition, Xn → X inL2(TM ) as n → ∞. If X ∈ L∞(TM ) in addition, this sequence can be constructedto be bounded in L∞(TM ).

Proof. Define Xk := 1|X|≤k X ∈ L2(TM ) ∩ L∞(TM ), k ∈ N, and, given anyt > 0, consider the element Xt,k := ttXk which, thanks to Proposition 6.24, belongsto D() ∩ L∞(TM ). By Theorem 6.23, we have Xt,k → ttX in H1,2(TM ) andXt,k → ttX in L2(TM ) as k → ∞ for every t > 0. Furthermore, ttX → X inH1,2(TM ) and ttX = ttX → X in L2(TM ) as t → 0 again by Theorem 6.23.The claim follows by a diagonal argument.

The following improvement of Proposition 6.24 is an instance of the correspon-dence between form domination and semigroup domination [57, 83, 89, 91]. Itextends analogous results for Riemannian manifolds without boundary [57, 58].

Theorem 6.26. For every X ∈ L2(TM ) and every t ≥ 0,

|ttX | ≤ pt|X | m-a.e.

Proof. Again, we restrict ourselves to t > 0. By the L2-continuity of both sides ofthe claimed inequality in X , it is sufficient to prove the latter for X ∈ Test(M ).Given any ε > 0, define the function ϕε ∈ C∞([0,∞)) ∩ Lip([0,∞)) by ϕε(r) :=(r+ε)1/2 −ε1/2. Moreover, let φ ∈ Test(M ) be nonnegative with ∆φ ∈ L∞(M ). Asin the proof of Proposition 6.24, one argues that the function Fε : [0, t] → R with

Fε(s) :=ˆ

M

φ pt−s(ϕε |tsX |2

)dm =

ˆ

M

pt−sφ[ϕε |tsX |2

]dm


is continuous on [0, t], locally absolutely continuous on (0, t), and in differentiatingit, integration and differentiation can be switched at L

1-a.e. s ∈ (0, t), yielding

F ′ε(s) = −

ˆ

M

∆pt−sφ[ϕε |tsX |2

]dm

+ 2ˆ

M

pt−sφ[ϕ′ε |tsX |2

]〈tsX,tsX〉 dm.

By Proposition 6.24, we have tsX ∈ L∞(TM ) and hence |tsX |2 ∈ F for everys ∈ (0, t) by Lemma 6.13. In particular ϕε |tsX |2, ϕ′

ε |tsX |2 ∈ F∩ L∞(M ), andhence by Proposition 2.11 and Remark 6.10,

F ′ε(s) =

ˆ

M

⟨∆pt−sφ,∇

[ϕε |tsX |2

]⟩dm

− 2ˆ

M

∇[pt−sφ

[ϕ′ε |tsX |2

]tsX

]: ∇tsX dm

=ˆ

M

[ϕ′ε |tsX |2

] ⟨∇pt−sφ,∇|tsX |2

⟩dm

− 2ˆ

M

pt−sφ[ϕ′′ε |tsX |2

]∇|tsX |2 ⊗ tsX : ∇tsX dm

− 2ˆ

M

[ϕ′ε |tsX |2

]∇pt−sφ⊗ tsX : ∇tsX dm

− 2ˆ

M


] ∣∣∇tsX∣∣2

HSdm

= −2ˆ

M

pt−sφ[ϕ′′ε |tsX |2

]∇|tsX |2 ⊗ tsX : ∇tsX dm

− 2ˆ

M


] ∣∣∇tsX∣∣2

HSdm.

In the last step, we used Proposition 6.11 to cancel out two integrals. Lastly, oneeasily verifies that −2r ϕ′′

ε (r) ≤ ϕ′ε(r) for every r ≥ 0, and that −ϕ′′

ε is nonnegative.Taking Lemma 6.13 into account, we thus get

− 2[ϕ′′ε |tsX |2

]∇|tsX |2 ⊗ tsX : ∇tsX

= −4[ϕ′′ε |tsX |2

]|tsX | ∇|tsX | ⊗ tsX : ∇tsX

∣∣2

≤ −4[ϕ′′ε |tsX |2

]|tsX |2

∣∣∇|tsX |∣∣ |∇tsX |HS

∣∣2

≤ 2[ϕ′ε |tsX |2

] ∣∣∇tsX∣∣2

HSm-a.e.

This shows that F ′(s) ≤ 0 for L1-a.e. s ∈ (0, t), whence

ˆ

M

φ[ϕε |ttX |2

]dm = Fε(t) ≤ Fε(0) =

ˆ

M

φ pt(ϕε |X |2) dm

for every ε > 0. Sending ε → 0 with the aid of Lebesgue’s theorem and using thearbitrariness of φ via Lemma 4.18 gives the desired assertion.

Note that the only essential tool to prove Proposition 6.24 and Theorem 6.26 isthe metric compatibility of ∇ from Proposition 6.11. In particular, no curvatureshows up in both statements.

Remark 6.27. In the notation of the proof of Theorem 6.26, Lemma 6.13 only guar-antees that |tsX | ∈ F, s ∈ (0, t). However, the required regularity |tsX |2 ∈ F

is unclear without any a priori information about L∞-L∞-regularizing propertiesof (tt)t≥0, which is precisely provided by Proposition 6.24. In turn, the proof ofthe latter only needs Greg-regularity of |tsX |2, which is true for any X ∈ L2(TM )

78 MATHIAS BRAUN

by Proposition 6.11. To integrate by parts in (6.8), this missing F-regularity iscompensated by Lemma 5.21, which is one key feature of the space Greg (recallRemark 5.22 as well).

Remark 6.28 (Heat kernel). Following the arguments for [16, Thm. 6.5], we deducefrom Theorem 6.26 that on any RCD(K,∞) space, K ∈ R — in fact, on any(tamed) Dirichlet space where (pt)t≥0 admits a heat kernel with Gaussian upperbounds as in (4.1) in [100], see also [5, Sec. 6.1] — (tt)t≥0 has a heat kernel in thesense of [16, Ch. 6]. The pointwise operator norm of the latter is m⊗2-a.e. no largerthan the heat kernel of (pt)t≥0, compare with [16, Thm. 6.7].

Corollary 6.29. The heat flow (tt)t≥0 uniquely extends to a semigroup of boundedlinear operators on Lp(TM ) for every p ∈ [1,∞] such that, for every X ∈ Lp(TM )and every t ≥ 0,

|ttX |p ≤ pt(|X |p

)m-a.e.,

and in particular

‖tt‖Lp(TM),Lp(TM) ≤ 1.

It is strongly continuous on Lp(TM ) if p < ∞ and weakly∗ continuous on L∞(TM ).

6.4. Bits of tensor calculus. In this section, we shortly outline basic elements ofgeneral nonsmooth tensor calculus. Fix r, s ∈ N0 throughout.

6.4.1. Tensor fields. Define the space of (L2-)tensor fields of type (r, s) over M by

L2(T srM ) := L2((T ∗)⊗sM ) ⊗ L2(T⊗rM ),

where all tensor products are intended in the sense of Subsection 1.4.5. For r = 0or s = 0, we employ the consistent interpretations

L2(T s0 M ) := L2((T ∗)⊗sM ),

L2(T 0r M ) := L2(T⊗rM ),

L2(T 00 M ) := L2(M ).

The L0-normed module induced by L2(T srM ) as in Subsection 1.4.3 is termedL0(T srM ). Given any T ∈ L0(T srM ), ω1, . . . , ωs ∈ L0(T ∗M ) as well as X1, . . . , Xr ∈L0(TM ), the pointwise scalar product of T and the element ω1 ⊗ · · · ⊗ ωs ⊗ X1 ⊗· · · ⊗ Xr ∈ L0(T srM ) is shortly written T (ω1, . . . , ωs, X1, . . . , Xr) ∈ L0(M ). (Thissection is the only place in our work where this bracket notation, strictly speaking,does not mean pointwise duality pairings, but rather pointwise pairings of elementsof the same vector space.)

6.4.2. Tensorial covariant derivative. We first introduce the concept of covariantderivative of suitable T ∈ L2(T rsM ). We start again with a motivating example.

Example 6.30. Suppose that M is a Riemannian manifold with boundary, andlet T ∈ Γ((T ∗)⊗rM ⊗ T⊗sM ) be an (r, s)-tensor field over M . Then, see e.g. [74,Ch. 8], ∇T ∈ Γ((T ∗)⊗rM ⊗ T⊗(s+1)M ) is the unique (r, s+1)-tensor field such thatfor every η1, . . . , ηr ∈ Γc(T ∗M ) and every Z, Y1, . . . , Ys ∈ Γc(TM ),

∇T (η1, . . . , ηr, Z, Y1, . . . , Ys) = d[T (η1, . . . , ηr, Y1, . . . , Ys)

](Z)

−r∑

i=1

T (∇Zηi) −s∑

j=1

T (∇ZYj)


on M . Here, we used the shorthand notations

T (∇Zηi) := T (η1, . . . , (∇Zη♯i )

︸︷︷︸i-th slot

, . . . , ηs, Y1, . . . , Yk),

T (∇ZYj) := T (η1, . . . , ηs, Y1, . . . ,∇ZYj︸︷︷︸(r+j)-th slot

, . . . , Yr)(6.9)

for i ∈ 1, . . . , r and j ∈ 1, . . . , s. As for the ordinary covariant derivative, seeExample 6.1 and (6.4) and also Remark 6.36 below, one thinks of ∇T (·, Z, ·) asdirectional derivative ∇ZT (·, ·).

As in Example 5.1 and Example 6.1, ∇T is still uniquely determined by theabove identity when requiring the latter only for η1, . . . , ηr and Z, Y1, . . . , Ys withvanishing normal parts at ∂M . Hence integration and integration by parts give

ˆ

M

∇T (η1, . . . , ηr, Z, Y1, . . . , Ys) dv

= −ˆ

M

T (η1, . . . , ηr, Y1, . . . , Ys) divZ dv

−ˆ

M

r∑

i=1

T (∇Zηi) dv −ˆ

M

s∑

j=1

T (∇ZYj) dv

by Example 3.9. Then ∇T is still uniquely defined by this identity.

Note that all relevant objects in Example 6.30, in particular the covariant deriva-tive of vector fields and directional derivatives, see Section 6.1 and Section 6.2, havealready been made sense of in our nonsmooth framework. Hence they can be usedto define the covariant derivative ∇T for appropriate T ∈ L2(T rsM ). Indeed, ther.h.s. of the above integral identity — with v replaced by m — makes sense for ar-bitrary such T , for η1, . . . , ηr ∈ Test(T ∗M ) and Z, Y1, . . . , Ys ∈ Test(TM ). ClearlyT (η1, . . . , ηr, Y1, . . . , Ys) ∈ L2(M ) and Z ∈ DTV(div) ∩ D(div) with divZ ∈ L2(M )and nZ = 0, which shows the well-definedness of the first integral. For the sec-ond, note that (∇Zη

♯i ) ∈ L2(T ∗M ), i ∈ 1, . . . , r, by (6.5), which directly yields

T (∇Zηi) ∈ L2(M ). The third integral is discussed analogously.This leads to the subsequent definition. In the sequel, we retain the shorthand

notations from (6.9).

Definition 6.31. We define the space W 1,2(T rsM ) to consist of all T ∈ L2(T rsM )for which there exists A ∈ L2(T rs+1M ) such that for every η1, . . . , ηr ∈ Test(T ∗M )and every Z, Y1, . . . , Ys ∈ Test(TM ),

ˆ

M

A(η1, . . . , ηr, Z, Y1, . . . , Ys) dm

= −ˆ

M

T (η1, . . . , ηr, Y1, . . . , Ys) divZ dm

−ˆ

M

r∑

i=1

T (∇Zηi) dm −ˆ

M

s∑

j=1

T (∇ZYj) dm.

In case of existence, the element A is unique, denoted by ∇T and termed the co-variant derivative of T .

The uniqueness follows by density of an appropriate class in L2(T rsM ), see (6.10)below and Subsection 1.4.5 for details. Clearly, W 1,2(T rsM ) is thus a vector space,and ∇ is a linear operator on it. Further properties of this covariant derivative aresummarized in Theorem 6.32 below. Thanks to Theorem 6.3 and (6.4), we haveW 1,2(T 0

1 M ) = W 1,2(TM ), and on these spaces, the notions of covariant derivative

80 MATHIAS BRAUN

from Definition 6.2 and Definition 6.31 coincide. Moreover, W 1,2(T 10 M ) coincides

with the image of W 1,2(TM ) under . Lastly, we have F⊂ W 1,2(T 00 M ), but in

general it seems hard to verify equality. See also Remark 7.3 below.We endow W 1,2(T rsM ) with the norm ‖ · ‖W 1,2(T r

s M) given by∥∥T

∥∥2

W 1,2(T rs M)

:=∥∥T

∥∥2

L2(T rs M)

+∥∥∇T

∥∥2

L2(T rs+1M)

.

We introduce the functional Ers : L2(T rsM ) → [0,∞] given by

Ers(T ) :=

ˆ

M

|∇T |2 dm if T ∈ W 1,2(T rsM ),

∞ otherwise.

The proof of the subsequent theorem follows completely similar lines as for inTheorem 5.3 and Theorem 6.3. We leave the details to the reader.

Theorem 6.32. The space W 1,2(T rsM ), the covariant derivative ∇ and the func-tional Ers have the following properties.

(i) W 1,2(T rsM ) is a separable Hilbert space w.r.t. ‖ · ‖W 1,2(T rs M).

(ii) The covariant derivative ∇ is a closed operator. That is, the image of themap Id × ∇ : W 1,2(T rsM ) → L2(T rsM ) × L2(T rs+1M ) is a closed subspace ofL2(T rs M ) × L2(T rs+1M ).

(iii) The functional Ers is L2-lower semicontinuous, and every T ∈ L2(T rsM )obeys the duality formula

Ers(T ) = sup

−2

n∑

k=1

ˆ

M

T (ηk1 , . . . , ηkr , Y

k1 , . . . , Y

ks ) div Y k0 dm

− 2n∑

k=1

ˆ

M

r∑

i=1

T (∇Y k0ηki ) dm − 2

n∑

k=1

ˆ

M

s∑

j=1

T (∇Y k0Y kj ) dm

−ˆ

M

∣∣∣n∑

k=1

ηk1 ⊗ · · · ⊗ ηkr ⊗ Y k0 ⊗ Y k1 ⊗ · · · ⊗ Y ks

∣∣∣2

dm :

n ∈ N, ηki ∈ Test(T ∗M ), Y kj ∈ Test(TM ).

n∑

j

6.4.3. Tensor algebra and Leibniz rule. Yet, unless r ∈ 0, 1 and s = 0 or r = 0and s ∈ 0, 1 we do not know whether W 1,2(T rsM ) is nontrivial. As we show inLemma 6.34, W 1,2(T rsM ) is in fact dense in L2(T rsM ) for arbitrary r, s ∈ N0, for itcontains the space of regular (r, s)-tensor fields given by

Reg(T rsM ) := n∑

k=1

ωk1 ⊗ · · · ⊗ ωkr ⊗Xk1 ⊗ · · · ⊗Xk

s :

n ∈ N, ωki ∈ Reg(T ∗M ), Xkj ∈ Reg(TM )

.

(6.10)

Since Test(M ) is both an algebra and closed under multiplication with constantfunctions, we consistently set Reg(T 0

0 M ) := Test(M ).As expected from the smooth setting [74, Ch. 8], the crucial tool to prove the

inclusion outlined above is the Leibniz rule — on every element of Reg(T rsM ), thecovariant derivative should pass through every slot. Technically, this requires todeal with the (L2-)tensor algebra TL2(M ), since every summand of (6.11) belowbelongs to a different L∞-tensor product.

To this aim, we consider the following sequence (Mn)n∈N0 of all possible finitetensor products of L2(T ∗M ) and L2(TM ) as in Subsection 1.4.5. Set

M0 := L2(M ),


M1 := L2(T ∗M ),

M2 := L2(TM ).

Inductively, if M2k−1, . . . ,M2(2k−1) are defined for a given k ∈ N then, for i ∈1, . . . , 2k+1, we set

M2(2k−1)+i :=

M2k−1+⌊i/2⌋ ⊗ L2(T ∗M ) if i is odd,M2k−1+⌊(i−1)/2⌋ ⊗ L2(TM ) otherwise.

Definition 6.33. The (L2)-tensor algebra over M is defined as

TL2 (M ) :=⊕

n∈N0

Mn.

All module operations, e.g. taking pointwise norms or multiplication with L∞-functions, can be made sense of componentwise for elements of TL2 (M ).

Lemma 6.34. We have the inclusion Reg(T sr ) ⊂ W 1,2(T rsM ). More precisely, forevery ω1, . . . , ωr ∈ Reg(T ∗M ) and every X1, . . . , Xs ∈ Reg(TM ), we have ω1 ⊗· · ·⊗ωr ⊗X1 ⊗ · · · ⊗Xs ∈ W 1,2(T rsM ) and, as an identity in TL2(M ),

∇(ω1 ⊗ · · · ⊗ ωr ⊗X1 ⊗ · · · ⊗Xs)

=r∑

i=1

ω1 ⊗ · · · ⊗ (∇ω♯i )︸︷︷︸i-th slot

⊗ · · · ⊗ ωr ⊗X1 ⊗ · · · ⊗Xs

+s∑

j=1

ω1 ⊗ · · · ⊗ ωr ⊗X1 ⊗ · · · ⊗ ∇Xj︸︷︷︸(r+j)-th slot

⊗ · · · ⊗Xs.

(6.11)

Proof. We first comment on the cases r ∈ 0, 1 and s = 0 or r = 0 and s ∈ 0, 1in which no formula (6.11) has to be shown. The case r = s = 0 is straightforwardfrom the identifications and inclusions Reg(T 0

0 M ) = Test(M ) ⊂ F⊂ W 1,2(T 00 M ) by

Lemma 3.15 and the Definition 3.5 of the L2-divergence. In this case, the covariantderivative and the gradient from Definition 3.4 agree. In the cases r = 0 and s = 1or r = 1 and s = 0, the claimed regularity follows from Theorem 6.3.

For r = s = 1, let ω ∈ Reg(T ∗M ) and X ∈ Reg(TM ), and let η ∈ Test(T ∗M )and Z, Y ∈ Test(TM ) be fixed. Then by (1.8), Lemma 3.15, Proposition 6.11, (6.4)and finally (1.10), we infer that

−ˆ

M

ω ⊗X(η, Y ) divZ dm

−ˆ

M

ω ⊗X((∇Zη♯), Y ) dm −

ˆ

M

ω ⊗X(η,∇ZY ) dm

=ˆ

M

d[〈ω, η〉〈X,Y 〉

](Z) dm

−ˆ

M

⟨ω, (∇Zη

♯)⟩

〈X,Y 〉 dm −ˆ

M

〈ω, η〉〈X,∇ZY 〉 dm

=ˆ

M

⟨(∇Zω

♯), η⟩

〈X,Y 〉 dm +ˆ

M

⟨ω, (∇Zη

♯)⟩

〈X,Y 〉 dm

+ˆ

M

〈ω, η〉〈∇ZX,Y 〉 dm +ˆ

M


−ˆ

M

⟨ω, (∇Zη

♯)⟩

〈X,Y 〉 dm −ˆ

M


=ˆ

M

(∇Zω♯) ⊗X(η, Y ) dm +

ˆ

M

ω ⊗ ∇ZX(η, Y ) dm.

82 MATHIAS BRAUN

By (6.4) and identification of the r.h.s. with the scalar product in L2(T 11 M ), it

follows that ω ⊗ X ∈ W 1,2(T 11 M ).

The case of general r, s ∈ N0 are now deduced similarly by induction over r ors while keeping the other variable fixed, respectively.

In the next final proposition, let us fix r′, s′ ∈ N0. Given any T ∈ L0(T rsM ) andS ∈ L0(T r

′

s′ M ), by T ⊠ S we mean the unique element of L0(T r+r′

s+s′ M ) such that forevery η1, . . . , ηr+r′ ∈ L0(T ∗M ) and every Y1, . . . , Ys+s′ ∈ L0(TM ),

T ⊠ S(η1, . . . , ηr+r′ , Y1, . . . , Ys+s′ )

= T (η1, . . . , ηr, Y1, . . . , Ys)

× S(ηr+1, . . . , ηr+r′ , Ys+1, . . . , Ys+s′ ) m-a.e.

Proposition 6.35 (Leibniz rule for tensor fields). Suppose that T ∈ W 1,2(T srM )and S ∈ Reg(T s

′

r′ M ). Then T ⊠ S ∈ W 1,2(T s+s′

r+r′ M ) and, as an identity in TL2 (M ),

∇(T ⊠ S) = ∇T ⊗ S + T ⊗ ∇S.Proof. We write S in the form

S := ωr+1 ⊗ · · · ⊗ ωr+r′ ⊗ Ys+1 ⊗ · · · ⊗ Ys+s′

for given ωr+1, . . . , ωr+r′ ∈ Reg(T ∗M ) and Ys+1, . . . , Ys+s′ ∈ Reg(TM ). Given anyη1, . . . , ηr+r′ ∈ Test(T ∗M ) and Z,X1, . . . , Xs+s′ ∈ Test(TM ), we abbreviate

f := T (η1, . . . , ηr, Y1, . . . , Ys),

g := S(ηr+1, . . . , ηr+r′ , Ys+1, . . . , Ys+s′)

and, for i ∈ r + 1, . . . , r + r′ and j ∈ s+ 1, . . . , s+ s′,

Ui := ωr+1 ⊗ · · · ⊗ (∇Zω♯i)

︸︷︷︸i-th slot

⊗ · · · ⊗ ωr+r′ ⊗ Ys+1 ⊗ · · · ⊗ Ys+s′ ,

Vj := ωr+1 ⊗ · · · ⊗ ωr+r′ ⊗ Ys+1 ⊗ · · · ⊗ ∇ZYj︸︷︷︸(r′+j)-th slot

⊗ · · · ⊗ Ys+s′ .

By (1.8), Proposition 6.11 and the Leibniz rule from Proposition 2.11, it followsthat g ∈ F∩ L∞(M ). By Lemma 3.15, we deduce that g Z ∈ DTV(div) ∩ D(div)with n(g Z) = 0 and that for every sequence (gn)n∈N in Test(M ) converging to gin F, we have div(gn Z) → div(g Z) in L2(M ) as n → ∞. By Lemma 3.15 againand the fact that gn Z ∈ Test(M ) for every n ∈ N,

−ˆ

M

T ⊠ S(η1, . . . , ηr+r′ , Y1, . . . , Ys+s′) divZ dm

−ˆ

M

r+r′∑

i=1

T ⊠ S(∇Zηi) dm −ˆ

M

s+s′∑

j=1

T ⊠ S(∇ZYj) dm

= − limn→∞

ˆ

M

f div(gn Z) dm +ˆ

M

f dg(Z) dm

− limn→∞

ˆ

M

gn

r∑

i=1

T (∇Zηi) dm − limn→∞

ˆ

M

gn

s∑

j=1

T (∇ZYj) dm

−ˆ

M

f

r+r′∑

i=r+1

S(∇Zηi) dm −ˆ

M

f

s+s′∑

j=s+1

S(∇Zηi) dm.

Applying Proposition 6.11 to the fifth last integral, then applying Lemma 6.34 tothe sum of those derivatives that fall on S, and finally using the definition of ∇T ,


the above sum is equal to

− limn→∞

ˆ

M

f div(gn Z) dm

+ˆ

M

f

r+r′∑

i=r+1

Ui(ηr+1, . . . , ηr+r′ , Ys+1, . . . , Ys+s′) dm

+ˆ

M

f

s+s′∑

j=s+1

Vj(ηr+1, . . . , ηr+r′ , Ys+1, . . . , Ys+s′) dm

− limn→∞

ˆ

M

gn

r∑

i=1

T (∇Zηi) dm − limn→∞

ˆ

M

gn

s∑

j=1

T (∇ZYj) dm

=ˆ

M

g∇T (η1, . . . , ηr, Z, Y1, . . . , Ys) dm

+ˆ

M

fr+r′∑

i=r+1

Ui(ηr+1, . . . , ηr+r′ , Ys+1, . . . , Ys+s′) dm

+ˆ

M

f

s+s′∑

j=s+1

Vj(ηr+1, . . . , ηr+r′ , Ys+1, . . . , Ys+s′) dm.

The claimed identity in TL2(M ) readily follows.

Remark 6.36. In a similar way as in (6.4), one can define the directional derivative∇ZT ∈ L0(T rs M ) of a given T ∈ W 1,2(T rsM ) in the direction of Z ∈ L0(TM ). Inthe notation of Proposition 6.35, given such Z the Leibniz rule becomes

∇Z(T ⊠ S) = ∇ZT ⊗ S + T ⊗ ∇ZS,

in TL2(M ), and accordingly in the framework of Lemma 6.34.

Remark 6.37. A more general Leibniz rule seems hard to obtain by evident integra-bility issues. Compare with Remark 7.9 below. One framework in which one couldinstead work is an appropriate version W 1,1(T r+r′

s+s′ M ) of Definition 6.31. However,as in the motivating remarks before Definition 5.28 and Definition 6.8, it is notclear if such a notion gives rise to nontrivial objects.

7. Exterior derivative

Throughout this chapter, let us fix k ∈ N0.

7.1. The Sobolev space D(dk). We now give a meaning to the exterior derivativeacting on suitable k-forms, i.e. elements of L2(ΛkT ∗M ) (recall Subsection 1.4.7).

7.1.1. Definition and basic properties. Before the motivating smooth Example 7.1,a notational comment is in order. Given ω ∈ L0(ΛkT ∗M ) and X0, . . . , Xk, Y ∈L0(TM ), we shall use the standard abbreviations

ω(Xi) := ω(X0, . . . , Xi, . . . , Xk)

X := ω(X0 ∧ · · · ∧Xi−1 ∧Xi+1 ∧ · · · ∧Xk),

ω(Y, Xi, Xj) := ω(Y,X0, . . . , Xi, . . . , Xj , . . . , Xk)

Xi := ω(Y ∧X0 ∧ · · · ∧Xi−1 ∧Xi+1 ∧ · · · ∧Xj−1 ∧Xj+1 ∧ · · · ∧Xk).

84 MATHIAS BRAUN

Example 7.1. On a Riemannian manifold M with boundary, the exterior derivatived: Γ(ΛkT ∗M ) → Γ(Λk+1T ∗M ) is defined by three axioms [74, Thm. 9.12]. It canbe shown [84, Sec. A.2] that the unique such d satisfies the following pointwise,chart-free representation for any ω ∈ Γ(ΛkT ∗M ) and any X0, . . . , Xk ∈ Γc(TM ):

dω(X0, . . . , Xk) =k∑

i=0

(−1)i d[ω(Xi)

](Xi)

+k∑

i=0

k∑

j=i+1

(−1)i+j ω([Xi, Xj ], Xi, Xj).

(7.1)

By the discussion from Section 1.2, the map d is still uniquely determined onΓ(ΛkT ∗M ) by this identity when restricting to those X0, . . . , Xk for which

〈X0, n〉 = · · · = 〈Xk, n〉 = 0 on ∂M .

In this case, integrating (7.1) leads toˆ

M

dω(X0, . . . , Xk) dv =ˆ

M

k∑

i=0

(−1)i ω(Xi) divXi dv

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j ω([Xi, Xj ], Xi, Xj) dv

after integration by parts in conjunction with Example 3.9. Given this integralidentity for every compactly supported X0, . . . , Xk ∈ Γ(ΛkT ∗M ) with vanishingnormal parts at ∂M as above, the differential dω ∈ Γ(Λk+1T ∗M ) of ω ∈ Γ(ΛkT ∗M )is of course still uniquely determined.

Now note that the r.h.s. of the last integral identity — with v replaced by m —is meaningful for arbitrary ω ∈ L2(ΛkT ∗M ) and X0, . . . , Xk ∈ Test(TM ). Indeed,ω(Xi) ∈ L2(M ) since X0, . . . , Xk ∈ L∞(TM ), and X0, . . . , Xk ∈ DTV(div)∩D(div)with divXi ∈ L2(M ) and nXi = 0 by Lemma 3.15, i ∈ 0, . . . , k. Moreover, by(6.5) the Lie bracket [Xi, Xj] belongs to L2(TM ), whence ω([Xi, Xj], Xi, Xj) ∈L2(M ), i ∈ 0, . . . , k and j ∈ i+ 1, . . . , k.

These considerations motivate the subsequent definition. (We only make explicitthe degree k in the name of the space, but not in the differential object itself.)

Definition 7.2. We define D(dk) to consist of all ω ∈ L2(ΛkT ∗M ) for which thereexists η ∈ L2(Λk+1T ∗M ) such that for every X0, . . . , Xk ∈ Test(TM ),ˆ

M

η(X0, . . . , Xk) dm =ˆ

M

k∑

i=0

(−1)i+1 ω(Xi) divXi dm

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j ω([Xi, Xj ], Xi, Xj) dm.

In case of existence, the element η is unique, denoted by dω and termed the exteriorderivative (or exterior differential) of ω.

The uniqueness follows by density of Test(Λk+1T ∗M ) in L2(Λk+1T ∗M ) as dis-cussed in Section 4.3. It is then clear that D(dk) is a real vector space and that dis a linear operator on it.

We always endow D(dk) with the norm ‖ · ‖D(dk) given by∥∥ω

∥∥2

D(dk):=

∥∥ω∥∥2

L2(ΛkT∗M)+

∥∥dω∥∥2

L2(Λk+1T∗M).


We introduce the functional Ed : L2(ΛkT ∗M ) → [0,∞] with

Ed(ω) :=

ˆ

M

|dω|2 dm if ω ∈ D(dk),

∞ otherwise.

We do not make explicit the dependency of Ed on the degree k. It will always beclear from the context which one is intended.

Remark 7.3. By Lemma 3.15 it is easy to see that F is contained in D(d0), andthat dω is simply the exterior differential from Definition 2.6, ω ∈ F. The reverseinclusion, however, seems more subtle, but at least holds true if M is intrinsicallycomplete as in Definition 4.16. Compare with Remark 5.22, [45, p. 136] and (theproof of) [45, Prop. 3.3.13].

Remark 7.4. Similarly to Remark 5.6 and Remark 6.4, motivated by its axiomati-zation in Riemannian geometry we expect the differential d to neither depend onconformal transformations of 〈·, ·〉, nor on drift transformations of m.

The next theorem collects basic properties of the above notions. It is proven ina similar fashion as Theorem 5.3 and Theorem 6.3.

Theorem 7.5. The space D(dk), the exterior derivative d and the functional Ed

satisfy the following properties.

(i) D(dk) is a separable Hilbert space w.r.t. ‖ · ‖D(dk).(ii) The exterior differential is a closed operator. That is, the image of the

map Id × d: D(dk) → L2(ΛkT ∗M ) × L2(Λk+1T ∗M ) is a closed subspace ofL2(ΛkT ∗M ) × L2(Λk+1T ∗M ).

(iii) The functional Ed is L2-lower semicontinuous, and for every ω ∈ L2(ΛkT ∗M )we have the duality formula

Ed(ω) = sup

2n∑

l=1

ˆ

M

k∑

i=0

(−1)i+1 ω(X li) divX l

i dm

+ 2n∑

l=1

ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j ω([X li , X

lj ], X

li , X

lj) dm

−ˆ

M

∣∣∣n∑

j=1

X l0 ∧ · · · ∧X l

k

∣∣∣2

dm : n ∈ N, X li ∈ Test(TM )

.

(iv) For every f0 ∈ Test(M ) ∪ R 1M and every f1, . . . , fk ∈ Test(M ) we havef0 df1 ∧ · · · ∧ dfk ∈ D(dk) with

d(f0 df1 ∧ · · · ∧ dfk) = df0 ∧ df1 ∧ · · · ∧ dfk,

with the usual interpretation d1M := 0. In particular Reg(ΛkT ∗M ) ⊂D(dk), and D(dk) is dense in L2(ΛkT ∗M ).

Proof. The items (i), (ii) and (iii) follow completely analogous lines as the proofs ofcorresponding statements in Theorem 5.3 and Theorem 6.3. We omit the details.

We turn to (iv). We concentrate on the proof of the claimed formula, fromwhich the last two statements then readily follow by linearity of d. First observethat the r.h.s. of the claimed identity belongs to L2(Λk+1T ∗M ). By definition(1.8) of the pointwise scalar product in L2(ΛkT ∗M ) and Proposition 6.24, we haveω(X1, . . . , Xk) ∈ F∩ L∞(M ), where ω := df1 ∧ · · · ∧ dfk. Direct computationsusing the Definition 6.18 of the Lie bracket and Theorem 6.3 yield

k∑

i=0

(−1)i d[ω(Xi)

](Xi) = −

k∑

i=0

k∑

j=i+1

(−1)i+j ω([Xi, Xj ], Xi, Xj) m-a.e.

86 MATHIAS BRAUN

For f0 ∈ Test(M ), it thus follows from Lemma 3.15 that

ˆ

M

k∑

i=0

(−1)i+1 f0 ω(Xi) divXi dm

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j f0 ω([Xi, Xj ], Xi, Xj) dm

=ˆ

M

k∑

i=0

(−1)i df0(Xi)ω(Xi) dm

+ˆ

M

k∑

i=0

(−1)i f0 d[ω(Xi)

](Xi) dm

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j f0 ω([Xi, Xj ], Xi, Xj) dm

=ˆ

M

(df0 ∧ ω)(X0, . . . , Xk) dm,

which shows the first claimed identity. The same computation can be done forf0 ∈ R 1M with the formal interpretation df0 := 0.

Remark 7.6. For arbitrary, not necessarily tamed Dirichlet spaces, certainly thespaces L2(ΛkT ∗M ) and L2(Λk+1T ∗M ) from Subsection 1.4.7 make sense. One isthen tempted to define the exterior derivative of f0 df1 ∧ · · · ∧ dfk for appropriatef0, . . . , fk ∈ Fe simply as df0 ∧ df1 ∧ · · · ∧ dfk ∈ L2(Λk+1T ∗M ). However, it is ingeneral not clear if d defined in that way is closable. In our approach, this is clearfrom Definition 7.2 by integration by parts, for which it has been crucial to knowthe existence of a large class of vector fields whose Lie bracket is well-defined. Inour approach, this is precisely Test(TM ), whose nontriviality — in fact, density inL2(TM ) — is a consequence of the (extended Kato condition on the) lower Riccibound κ, see Subsection 4.2.1 and Section 4.3.

7.1.2. Calculus rules. We proceed with further calculus rules for d. In view ofProposition 7.8 below, the following preliminary lemma is required.

Lemma 7.7. Suppose that ω ∈ D(dk), and that f ∈ F∩ L∞(M ). Then for everyX0, . . . , Xk ∈ Test(TM ),

ˆ

M

f dω(X0, . . . , Xk) dm

=ˆ

M

k∑

i=0

(−1)i+1 ω(Xi) div(f Xi) dm

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j f ω([Xi, Xj ], Xi, Xj) dm.

Proof. We first prove the claim for f ∈ Test(M ). As f X0 ∈ Test(M ), by definitionof the Lie bracket and Lemma 6.9 we have

[f X0, Xj ] = ∇f X0Xj − ∇Xj(f X0)

= f ∇X0Xj − df(Xj)X0 − f ∇XjX0

= f [X0, Xj] − df(Xj)X0


for every j ∈ 1, . . . , k. Hence, by Lemma 3.13,ˆ

M

f dω(X0, . . . , Xk) dm

=ˆ

M

dω(f X0, X1, . . . , Xk) dm

= −ˆ

M

ω(X0) df(X0) dm +ˆ

M

k∑

i=0

(−1)i+1f ω(Xi) divXi dm

+ˆ

M

k∑

i=0

k∑

j=i+1

f ω([Xi, Xj ], Xi, Xj) dm

+ˆ

M

k∑

j=1

(−1)j+1 ω(Xj) df(Xj) dm.

Since div(f Xi) = df(Xi) + f divXi m-a.e. by Lemma 3.13, we are done.The claim for general f ∈ F∩ L∞(M ) follows by the approximation result from

Lemma 4.18 together with Lemma 3.13.

Proposition 7.8 (Leibniz rule). Let ω ∈ D(dk) and, for some k′ ∈ N0, supposethat ω′ ∈ Reg(Λk

′

T ∗M ). Then ω ∧ ω′ ∈ D(dk+k′

) with

d(ω ∧ ω′) = dω ∧ ω′ + (−1)k ω ∧ dω′.

Proof. We proceed by induction on k′ and start with k′ = 0. In this case, ω′ is sim-ply an element f ∈ Test(M ). Given any X0, . . . , Xk ∈ Test(TM ), by Lemma 3.13and Lemma 7.7 we obtainˆ

M

k∑

i=0

(−1)i+1 f ω(Xi) divXi dm

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j f ω([Xi, Xj], Xi, Xj) dm

=ˆ

M

k∑

i=0

(−1)i+1 ω(Xi) div(f Xi) dm +ˆ

M

k∑

i=0

(−1)i ω(Xi) df(Xi) dm

+ˆ

M

k∑

i=0

k∑

j=i+1

(−1)i+j f ω([Xi, Xj], Xi, Xj) dm

=ˆ

M

f dω(X0, . . . , Xk) dm +ˆ

M

(df ∧ ω)(X0, . . . , Xk) dm.

In the last equality, we used the definition (1.15) of the pointwise scalar product inL2(ΛkT ∗M ). Therefore, we obtain that f ω ∈ D(dk) with

d(f ω) = f dω + df ∧ ω = f dω + (−1)k ω ∧ df,

which is precisely the claim for k′ = 0.Before we proceed with the induction step, we show the claim under the assump-

tion that ω′ := df for some f ∈ Test(M ), in which case we more precisely claimthat ω ∧ df ∈ D(dk+1) with

d(ω ∧ df) = dω ∧ df, (7.2)

keeping in mind that d(df) = 0 by Theorem 7.5. To this aim, let X0, . . . , Xk+1 ∈Test(TM ). By definition (1.15) of the pointwise scalar product and Lemma 7.7 —

88 MATHIAS BRAUN

which can be applied since df(Xi) ∈ F∩ L∞(M ) by Proposition 6.11 — we getˆ

M

(dω ∧ df)(X0, . . . , Xk+1) dm

=ˆ

M

k+1∑

i=0

(−1)i+k+1 dω(Xi) df(Xi) dm

=ˆ

M

k+1∑

i=0

k+1∑

j=0,j 6=i

aij ω(Xi, Xj) div[df(Xi)Xj

]dm

+ˆ

M

k+1∑

i=0

k+1∑

j=0,j 6=i

k+1∑

j′=j+1,j′ 6=i

bijj′ ω([Xj , Xj′ ], Xi, Xj , Xj′) df(Xi) dm,

where, for i, j, j′ ∈ 0, . . . , k + 1,

aij :=

(−1)i+j+k+1 if j ≤ i,

(−1)i+j+k otherwise,

bijj′ :=

(−1)i+j+j

′+k if j < i < j′,

(−1)i+j+j′+k+1 otherwise.

Then (7.2) directly follows since, by Lemma 6.16,

div[df(Xi)Xj

]− div

[df(Xj)Xi

]

= df(Xi) divXj − df(Xj) divXi − df([Xi, Xj ]) m-a.e.

Now we are ready to perform the induction step. Given the assertion for k′ − 1with k′ ∈ N, by linearity it suffices to consider the case ω′ := f0 df1 ∧ · · · ∧ dfk′ ,where f0 ∈ Test(TM ) ∪ R 1M and f1, . . . , fk′ ∈ Test(M ). By Theorem 7.5 we havef0 ω ∈ D(dk). Writing ω′′ := df2 ∧ · · · ∧ dfk′ thus yields

d(ω ∧ ω′) = d[(ω ∧ f0 df1) ∧ ω′′

]

= d[(f0 ω) ∧ df1

]∧ ω′′

=[f0 dω + df0 ∧ ω

]∧ df1 ∧ ω′′

= dω ∧ ω′ + (−1)k ω ∧ dω′

where we used the induction hypothesis in the second identity and Theorem 7.5 inthe last two equalities.

Remark 7.9. In Proposition 7.8, by evident integrability issues we cannot go re-ally beyond the assumption ω ∈ Reg(Λk

′

T ∗M ), not even to the space Dreg(dk)introduced in the subsequent Definition 7.10.

Definition 7.10. We define the space Dreg(dk) ⊂ D(dk) by

Dreg(dk) := cl‖·‖D(d)Reg(ΛkT ∗M ).

This definition is non-void thanks to Theorem 7.5 — in fact, Dreg(dk) is a densesubspace of L2(ΛkT ∗M ). As in Remark 6.7 we see that Dreg(dk) coincides withthe closure of Test(ΛkT ∗M ) in D(dk) if M is intrinsically complete, but might belarger in general. In line with this observation, we also do not know if D(d0) = F

unless constant functions belong to F.

Proposition 7.11. For every ω ∈ Dreg(dk), we have dω ∈ Dreg(dk+1) with

d(dω) = 0.


Proof. The statement for ω ∈ Reg(ΛkT ∗M ) follows from Theorem 7.5 above. Bydefinition of D(dk) and the closedness of d again by Theorem 7.5, the claim extendsto arbitrary ω ∈ D(dk).

Remark 7.12. A locality property for d such as

1ω=0 dω = 0

for general ω ∈ D(dk) seems hard to obtain from our axiomatization. Comparewith a similar remark at [45, p. 140].

7.2. Nonsmooth de Rham cohomology and Hodge theorem. Motivated byProposition 7.11, the goal of this section is to make sense of a nonsmooth de Rhamcomplex. The link with the smooth setting is outlined in Remark 7.25.

Given k ∈ N0 we exceptionally designate by dk the exterior differential definedon D(dk), which is well-defined by Proposition 7.11. Define the spaces Ck(M ) andEk(M ) of closed and exact k-forms by

Ck(M ) := ker dk

: =ω ∈ D(dk) : dω = 0

,

Ek(M ) := im dk−1

: =ω ∈ D(dk) : ω = dω′ for some ω′ ∈ D(dk−1)

.

By Theorem 7.5, we know that Ck(M ) is a closed subspace of L2(ΛkT ∗M ), but notif the same is true for Ek(M ). Since Ek(M ) ⊂ Ck(M ) and Ck(M ) is L2-closed, theL2-closure of Ek(M ) is contained in Ck(M ) as well, hence the following definitionis meaningful and non-void.

Definition 7.13. The k-th de Rham cohomology group of M is defined by

HkdR(M ) := Ck(M )

/cl‖·‖

L2(ΛkT ∗M)Ek(M ).

In view of the Hodge Theorem 7.23, we first need to make sense of the HodgeLaplacian and of harmonic k-forms. We start with the following. (Again, we onlymake explicit the degree k in the denotation of the space, but not of the differentialobject itself.)

Definition 7.14. Given any k ≥ 1, the space D(δk) is defined to consist of allω ∈ L2(ΛkT ∗M ) for which there exists ρ ∈ L2(Λk−1T ∗M ) such that for every η ∈Test(Λk−1T ∗M ), we have

ˆ

M

〈ρ, η〉 dm =ˆ

M

〈ω, dη〉 dm.

If it exists, ρ is unique, denoted by δω and called the codifferential of ω. We simplydefine D(δ0) := L2(M ) and δ := 0 on this space.

By the density of Test(Λk−1T ∗M ) in L2(Λk−1T ∗M ), the uniqueness statementis indeed true. Furthermore, δ is a closed operator, i.e. the image of the as-signment Id × δ : D(δk) → L2(ΛkT ∗M ) × L2(Λk−1T ∗M ) is closed in L2(ΛkT ∗M ) ×L2(Λk−1T ∗M ). Lastly, by comparison of Definition 7.14 with Definition 3.5 we haveD(δ1) = D(div), and for every ω ∈ D(δ1),

δω = − divω♯ m-a.e. (7.3)

The next result shows that D(δk) is nonempty — in fact, it is dense in L2(ΛkT ∗M ).There and in the sequel, for appropriate f1, . . . , fk ∈ Fe and i, j ∈ 1, . . . , k withi < j, we use the abbreviations

df i := df1 ∧ · · · ∧ df i ∧ · · · ∧ dfk

90 MATHIAS BRAUN

df i, df j := df1 ∧ · · · ∧ df i ∧ · · · ∧ df j ∧ · · · ∧ dfk.

Lemma 7.15. For every f0 ∈ Test(M ) ∪ R 1M and every f1, . . . , fk ∈ Test(M ),we have f0 df1 ∧ · · · ∧ dfk ∈ D(δk) with

δ(f0 df1 ∧ · · · ∧ dfk) =k∑

i=1

(−1)i[f0 ∆fi + 〈df0, dfi〉

]df i

+k∑

i=1

k∑

j=i+1

(−1)i+j f0 [∇fi,∇fj ] ∧ df i, df j,

with the usual interpretation d1M := 0.

Proof. We abbreviate ω := f0 df1 ∧ · · · ∧ dfk. By linearity, it clearly suffices toconsider the defining property from Definition 7.14 for η := g1 dg2 ∧ · · · ∧ dgk,g1, . . . , gk ∈ Test(M ). Write Sk for the set of permutations of 1, . . . , k. Then by(1.15) and the Leibniz formula,

ˆ

M

〈ω, dη〉 dm =ˆ

M

∑

σ∈Sk

sgnσ f0

⟨∇g1,∇fσ(1)

⟩ k∏

i=2

⟨∇gi,∇fσ(i)

⟩dm.

Since ∇g1 ∈ D(div), regardless of whether f0 ∈ Test(M ) or f0 ∈ R 1M — and withappropriate interpretation ∇f0 := 0 in the latter case — integration by parts andthen using Lemma 3.15 and Proposition 6.11 yields

ˆ

M

〈ω, dη〉 dm = −ˆ

M

∑

σ∈Sk

sgnσ g1 div[f0 ∇fσ(1)

k∏

i=2

⟨∇gi,∇fσ(i)

⟩]dm

= −ˆ

M

∑

σ∈Sk

sgnσ g1

⟨∇f0,∇fσ(1)

⟩ k∏

i=2

⟨∇gi,∇fσ(i)

⟩dm

−ˆ

M

∑

σ∈Sk

sgnσ g1 f0 ∆fσ(1)

k∏

i=2

⟨∇gi ∇fσ(i)

⟩dm

−ˆ

M

∑

σ∈Sk

sgnσ g1 f0

k∑

i=2

[Hess gi(∇fσ(1),∇fσ(i))

×k∏

j=2,j 6=i

⟨∇gj ,∇fσ(j)

⟩]dm

−ˆ

M

∑

σ∈Sk

sgnσ g1 f0

k∑

i=2

[Hess fσ(i)(∇fσ(1),∇gi)

×k∏

j=2,j 6=i

⟨∇gj ,∇fσ(j)

⟩]dm.

The second last integral vanishes identically, which follows by symmetry of theHessian and by comparing a given σ ∈ Sk with the permutation that swaps σ(1)and σ(i) in σ, i ∈ 2, . . . , k. The claim follows from the combinatorial formulas

∑

σ∈Sk,σ(1)=I

sgnσk∏

j=1

⟨∇gj ,∇fσ(j)

⟩= (−1)I+1

⟨dg2 ∧ · · · ∧ dgk, dfI

⟩


∑

σ∈Sk,σ(1)=I,σ(J)=K

sgnσk∏

j=1,j 6=J

⟨∇gi,∇fσ(i)

⟩= aIJK

⟨dgJ, dfI , dfK

⟩

for every I,K ∈ 1, . . . , k with I 6= K and every J ∈ 2, . . . , k, where

aIJK :=

(−1)1+I+J+K if I < K,

(−1)I+J+K otherwise,

from the identity

Hess fK(∇fI , ·) − Hess fI(∇fK , ·) = [∇fI ,∇fK ]

stemming from the definition of the Lie bracket and Theorem 5.3, andk∑

J=2

(−1)J⟨[∇fI ,∇fK ],∇gJ

⟩ ⟨dgJ, dfI , dfK

⟩

=⟨dg2 ∧ · · · ∧ dgk, [∇fI ,∇fK ] ∧ dfI , dfK

⟩.

Remark 7.16. By approximation, using Theorem 7.5 and Lemma 7.15, one readilyproves the following. Let f ∈ Feb and ω ∈ H1,2(T ∗M ). Assume furthermore thatdf ∈ L∞(T ∗M ) or that ω ∈ L∞(T ∗M ). Then f ω ∈ H1,2(T ∗M ) with

d(f ω) = f dω + df ∧ ω,

δ(f ω) = f δω − 〈df, ω〉.

Definition 7.17. We define the space W 1,2(ΛkT ∗M ) by

W 1,2(ΛkT ∗M ) := D(dk) ∩ D(δk).

By Theorem 7.5 and Lemma 7.15, we already know that W 1,2(ΛkT ∗M ) is adense subspace of L2(ΛkT ∗M ).

We endow W 1,2(ΛkT ∗M ) with the norm ‖ · ‖W 1,2(ΛkT∗M) given by

‖ω‖2W 1,2(ΛkT∗M) := ‖ω‖2

L2(ΛkT∗M) + ‖dω‖2L2(Λk+1T∗M) + ‖δω‖2

L2(Λk−1T∗M)

and we define the contravariant functional Econ : L2(ΛkT ∗M ) → [0,∞] by

Econ(ω) :=

12

ˆ

M

[|dω|2 + |δω|2

]dm if ω ∈ W 1,2(ΛkT ∗M ),

∞ otherwise.

Arguing as for Theorem 5.3, Theorem 6.3 and Theorem 7.5, W 1,2(ΛkT ∗M ) be-comes a separable Hilbert space w.r.t. ‖ · ‖W 1,2(ΛkT∗M). Moreover, the functionalEcon is clearly L2-lower semicontinuous.

Again by Theorem 7.5 and Lemma 7.15, we have Reg(ΛkT ∗M ) ⊂ W 1,2(ΛkT ∗M ),so that the following definition makes sense.

Definition 7.18. The space H1,2(ΛkT ∗M ) ⊂ W 1,2(ΛkT ∗M ) is defined by

H1,2(ΛkT ∗M ) := cl‖·‖W 1,2(ΛkT ∗M)

Reg(ΛkT ∗M ).

Remark 7.19 (Absolute boundary conditions). We adopt the interpretation thatω and dω have “vanishing normal components” for any given ω ∈ H1,2(ΛkT ∗M ).For instance, on a compact Riemannian manifold M with boundary, by (1.3) everyω ∈ Reg(ΛkT ∗M ) satisfies the absolute boundary conditions [88, Sec. 2.6]

nω = 0,

n dω = 0(7.4)

92 MATHIAS BRAUN

s-a.e. at ∂M . By Gaffney’s inequality — see Remark 7.20 below — Proposition 7.11and Lemma 7.15, both ω and dω belong to the corresponding W 1,2-Sobolev spacesover F := ΛkT ∗M and F := Λk+1T ∗M induced by the Levi-Civita connection ∇ asintroduced in Section 1.2, respectively, and these inclusions are continuous. Hence,the trace theorem can be applied, and (7.4) passes to the limit in the definition ofH1,2(ΛkT ∗M ). In particular, (7.4) holds s-a.e. for every ω ∈ H1,2(ΛkT ∗M ).

Remark 7.20 (Gaffney’s inequality). In general, H1,2(ΛkT ∗M ) does not coincidewith W 1,2(ΛkT ∗M ). On a compact Riemannian manifold M with boundary, ourusual argument using Proposition 1.2 gives the claim, up to an important technicaldetail to be fixed before. (For notational simplicity, we restrict ourselves to thecase k = 1. In the general case, H1,2(TM ) has to be replaced by W 1,2(F) definedaccording to Section 1.2 with F := ΛkT ∗M .)

In general, d and δ are continuous w.r.t. convergence in H1,2(TM ) [sic] by [88,p. 62], whence H1,2(T ∗M ) ⊂ H1,2(TM ) with continuous inclusion. However, toapply the trace theorem to infer that all elements of H1,2(T ∗M ) have vanishingnormal component at ∂M , the reverse inclusion is required (compare with theforegoing Remark 7.19 and with Lemma 8.8 below). The latter is a classical resultby Gaffney, see [88, Cor. 2.1.6] for a proof: there exists a finite constant C > 0 suchthat for every ω ∈ H1,2(TM ) with nω = 0 s-a.e. on ∂M , we have

∥∥ω♯∥∥2

W 1,2(TM)≤ C

∥∥ω∥∥2

W 1,2(T∗M).

Definition 7.21. The space D(~∆k) is defined to consist of all ω ∈ H1,2(ΛkT ∗M )for which there exists α ∈ L2(ΛkT ∗M ) such that for every η ∈ H1,2(ΛkT ∗M ),

ˆ

M

〈α, η〉 dm =ˆ

M

[〈dω, dη〉 + 〈δω, δη〉

]dm.

In case of existence, the element α is unique, denoted by ~∆kω and termed the HodgeLaplacian of ω. Moreover, the space H(ΛkT ∗M ) of harmonic k-forms is defined asthe space of all ω ∈ H1,2(ΛkT ∗M ) with dω = 0 and δω = 0.

For the most important case k = 1, we shall write ~∆ instead of ~∆k. By (1.5),~∆0 = −∆.

Moreover, the Hodge Laplacian ~∆k is a closed operator, which can be e.g. seen byidentifying ~∆kω, ω ∈ D(~∆k), with the only element in the subdifferential of Econ(ω),where the functional Econ is defined in (8.16) below [45, p. 145]. In particular,H(ΛkT ∗M ) is a closed subspace of L2(ΛkT ∗M ).

Remark 7.22. Our definition of harmonic k-forms follows [88, Def. 2.2.1]. Of course,in the framework of Definition 7.21, ω ∈ H(ΛkT ∗M ) if and only if ω ∈ D(~∆k) with~∆kω = 0, see e.g. [45, p. 145]. This, however, is a somewhat implicit consequence ofthe interpretation of any ω ∈ H(ΛkT ∗M ) as obeying absolute boundary conditions(recall Remark 7.19). Compare with [88, Prop. 1.2.6, Prop. 2.1.2, Cor. 2.1.4]. Ingeneral, the vanishing of ~∆kω is a weaker condition than asking for dω and δω tovanish identically for appropriate ω ∈ L2(ΛkT ∗M ) [88, p. 68].

We can now state and prove the following variant of Hodge’s theorem.

Theorem 7.23 (Hodge theorem). The map ω 7→ [ω] from H(ΛkT ∗M ) into HkdR(M )

is an isomorphism of Hilbert spaces.

Proof. Set H := Ck(M ) and V := Ek(M ). Thanks to Theorem 7.5, H is a Hilbertspace w.r.t. ‖ · ‖H := ‖ · ‖L2(ΛkT∗M). Moreover, V is a subspace of H . Lastly, byDefinition 7.14 it is elementary to see that

V ⊥ = H(ΛkT ∗M ),


where we intend the orthogonal complement w.r.t. the usual scalar product inL2(ΛkT ∗M ). Therefore, Theorem 7.23 follows since by basic Hilbert space the-ory [69, Thm. 2.2.3], the map sending any ω ∈ V ⊥ to ω + cl‖·‖H

V ∈ H/cl‖·‖HV is

a Hilbert space isomorphism.

Remark 7.24. It is unclear if W 1,2(ΛkT ∗M ) \H1,2(ΛkT ∗M ) contains elements withvanishing differential and codifferential, hence our choice of the domain of definitionof ~∆k and of harmonic k-forms. Compare with [45, Rem. 3.5.16].

Remark 7.25. Theorem 7.23 is a variant of the Hodge theorem on compact man-ifolds M with boundary [88, Thm. 2.6.1]. Quite interestingly, in contrast to oursetting — where boundary conditions somewhat come as a byproduct of our class of“smooth k-forms” — no boundary conditions are needed to build the correspondingde Rham cohomology group Hk

dR(M ) [88, p. 103]. Still, the latter is isomorphic tothe space of harmonic Neumann fields [88, Def. 2.2.1] which by definition satisfyabsolute boundary conditions. This follows from the Hodge–Morrey decomposition[88, Thm. 2.4.2] combined with the Friedrichs decomposition [88, Thm. 2.4.8].

It is worth mentioning that in this setting, dimH(ΛkT ∗M ) coincides with thek-th Betti number of M [88, p. 68].

8. Curvature measures

We are now in a position to introduce various concepts of curvature. In Section 8.1,we prove our main result, Theorem 8.9, where the κ-Ricci measure Ricκ is madesense of, among others in terms of ∆2κ. In Section 8.2, we separate κ from ∆2κ,which induces the measure-valued Laplacian ∆ fully compatible with the definitionof the measure-valued divergence div, see Definition 8.11. In turn, this allows usto define a Ricci curvature and an intrinsic second fundamental form on M .

8.1. κ-Ricci measure. Our main Theorem 8.9 requires some technical preliminaryingredients that are subsequently discussed.

8.1.1. Preliminary preparations.

Lemma 8.1. For every g ∈ Test(M ) ∪ R 1M and every f ∈ Test(M ), we haveg df ∈ D(~∆) with

~∆(g df) = −g d∆f − ∆g df − 2 Hess f(∇g, ·),with the usual interpretations ∇1M := 0 and ∆1M := 0. More generally, for everyX ∈ Reg(TM ) and every h ∈ Test(M ) ∪ R 1M , we have hX ∈ D(~∆) with

~∆(hX) = h ~∆X − ∆hX − 2 ∇∇hX.

Proof. The respective r.h.s.’s of the claimed identities belong to L2(T ∗M ), whichgrants their meaningfulness.

To prove the first, we claim that for every f ′, g′ ∈ Test(M ),ˆ

M

⟨d(g df), d(g′ df ′)

⟩dm +

ˆ

M

δ(g df) δ(g′ df ′) dm

= −ˆ

M

⟨g d∆f + ∆g df + 2 Hess f(∇g, ·), g′ df ′

⟩dm.

By linearity of both sides in g′ df ′ and density, the first identity for ~∆(g df) thenreadily follows. Indeed, since δ(g df) = −〈∇g,∇f〉 − g∆f m-a.e. by (7.3) — withappropriate interpretation if g ∈ R 1M according to (3.2) — δ(g df) belongs to F

by Proposition 6.11, and we haveˆ

M

δ(g df) δ(g′ df ′) dm

94 MATHIAS BRAUN

= −ˆ

M

⟨d[〈∇g,∇f〉 + g∆f

], g′ df ′

⟩dm

= −ˆ

M

⟨Hess g(∇f, ·) + Hess f(∇g, ·) + dg∆f + g d∆f, g′ df ′

⟩dm,

with Hess g := 0 whenever g ∈ R 1M . On the other hand, by Theorem 7.5 we haved(g df) = dg ∧ df , which belongs to D(δ2) by Lemma 7.15 with

ˆ

M

⟨d(g df), d(g′ df ′)

⟩dm

=ˆ

M

⟨δ(dg ∧ df), g′ df ′

⟩dm

=ˆ

M

⟨∆f dg − ∆g df − [∇g,∇f ], g′ df ′

⟩dm.

Adding up these two identities yields the claim since

[∇g,∇f ] = Hess f(∇g, ·) − Hess g(∇f, ·).Concerning the second claim, from what we already proved, the linearity of ~∆

and since Test(M ) ∪ R 1M is an algebra, it follows that hX ∈ D(~∆). Again bylinearity, it thus suffices to consider the case X := g∇f , g ∈ Test(M ) ∪ R 1M andf ∈ Test(M ). Indeed,

~∆(h g df) = −h g df − ∆(h g) df − 2 Hess f(∇(h g), ·)= −h g d∆f − g∆h df − 2〈∇h,∇g〉 df − h∆g df

− 2 gHess f(∇h, ·) − 2 hHess f(∇g, ·)= h ~∆(g df) − ∆h (g df) − 2

[∇∇h(g∇f)

]♯.

In the last identity, we used the identity

∇(g∇f) = ∇g ⊗ ∇f + g (Hess f)♯

inherited from Theorem 5.3.

Recall from Lemma 2.7 that ∇ Test(M ), a set consisting of m-essentially boundedelements, generates L2(TM ) in the sense of L∞-modules. Hence, for every A ∈L2(T⊗2M ), by (1.12) its symmetric part obeys the duality formula

∣∣Asym

∣∣2

HS= esssup

2A :

m∑

j=1

∇hj ⊗ ∇hj −∣∣∣m∑

j=1

∇hj ⊗ ∇hj∣∣∣2

HS:

m ∈ N, h1, . . . , hm ∈ Test(M ).

(8.1)

This is crucial in the next Lemma 8.2. Therein, the divergence of X ∈ Reg(TM ) isunderstood in the sense of Definition 3.5. Recall that if X ∈ Test(TM ), this is thesame as interpreting it according to Definition 3.7 by Lemma 3.11.

Lemma 8.2. For every X ∈ Reg(TM ), we have |X |2 ∈ D(∆2κ) with

∆2κ |X |22

≥[∣∣∇X

∣∣2

HS−

⟨X, (~∆X)♯

⟩]m.

Proof. The Leibniz rule for ∆2κ [41, Cor. 6.3] together with polarization and thelinearity of ∆2κ on F, recall in particular Proposition 4.20, ensure that |X |2 ∈ F,and in fact |X |2 ∈ D(∆2κ).

Write X := g1 ∇f1 + · · · + gn ∇fn for certain gi ∈ Test(M ) ∪ R 1M and fi ∈Test(M ), i ∈ 1, . . . , n. Retaining the notation from Lemma 5.9 for N ′ := ∞, wefirst claim the identity

ρ1[f, g] = ∆2κ |X |22

+[⟨X, (~∆X)♯

⟩−

∣∣(∇X)asym

∣∣2

HS

]m. (8.2)


Again by the Leibniz rule for ∆2κ from [41, Cor. 6.3] and Proposition 5.32,

∆2κ |X |22

=12

n∑

i,i′=1

gi gi′ ∆2κ〈∇fi,∇fi′〉 +n∑

i,i′=1

⟨∇[gi gi′ ],∇〈∇fi,∇fi′〉

⟩m

+12

n∑

i,i′=1

〈∇fi,∇fi′〉 ∆[gi gi′ ]m

=12

n∑

i,i′=1

gi gi′ ∆2κ〈∇fi,∇fi′〉

+n∑

i,i′=1

[Hess fi(∇fi′ ,∇[gi gi′ ]) + Hess fi′(∇fi,∇[gi gi′ ])

]m

+n∑

i,i′=1

[gi′ ∆gi 〈∇fi,∇fi′〉 + 〈∇gi,∇gi′〉〈∇fi,∇fi′〉

]m

=12

n∑

i,i′=1

gi gi′ ∆2κ〈∇fi,∇fi′〉

+ 2n∑

i,i′=1

[gi Hess fi(∇fi′ ,∇gi′) + gi Hess fi′(∇fi,∇gi′)

]m

+n∑

i,i′=1

[gi′ ∆gi 〈∇fi,∇fi′〉 + 〈∇gi,∇gi′〉〈∇fi,∇fi′〉

]m.

Furthermore, by Lemma 8.1 we obtain

~∆X = −n∑

i=1

[gi d∆fi + ∆gi dfi + 2 Hess fi(∇gi, ·)

],

which entails⟨X, (~∆X)♯

⟩= −

n∑

i,i′=1

[gi′ ∆gi 〈∇fi,∇fi′〉 + 2 gi′ Hess fi(∇gi,∇fi′)

]

−n∑

i,i′=1

[gi gi′ 〈∇∆fi,∇fi′〉

]m-a.e.

Next, by item (iii) in Theorem 6.3, the symmetry of the Hessian by Theorem 5.3,and the definition of the anti-symmetric part of an element in L2(T⊗2M ),

(∇X)asym =12

n∑

i=1

[∇gi ⊗ ∇fi − ∇fi ⊗ ∇gi

]

from which we get

∣∣(∇X)asym

∣∣2

HS=

12

n∑

i,i′=1

[〈∇fi,∇fi′〉〈∇gi,∇gi′〉 − 〈∇fi,∇gi′〉〈∇gi,∇fi′〉

]m-a.e.

Lastly, the definition (4.3) of the measure-valued Γ2-operator yields

Γ2κ2 (fi, fi′) =

12

∆2κ〈∇fi,∇fi′〉 − 12

[〈∇∆fi,∇fi′〉 + 〈∇∆fi′ ,∇fi〉

]m

for every i, i′ ∈ 1, . . . , n. Patching terms together straightforwardly leads to (8.2).Therefore, w.r.t. m we have

[∆2κ |X |2

2

]⊥

= ρ1[f, g]⊥ ≥ 0

96 MATHIAS BRAUN

thanks to Lemma 5.9. To finally prove the claimed inequality, it thus suffices toshow that, setting δ2κ|X |2/2 := d(∆2κ|X |2/2)≪/dm,

δ2κ |X |22

≥∣∣∇X

∣∣2

HS−

⟨X, (~∆X)♯

⟩m-a.e. (8.3)

Indeed, having (8.2) at our disposal, Lemma 5.9 implies that for every m ∈ N andfor every h1, . . . , hm ∈ Test(M ),

∣∣∣∇X :m∑

j=1

∇hj ⊗ ∇hj∣∣∣ ≤

[δ2κ |X |2

2+

⟨X, (~∆X)♯

⟩−

∣∣(∇X)asym

∣∣2

HS

]1/2

×∣∣∣m∑

j=1

∇hj ⊗ ∇hj∣∣∣HS

m-a.e.

Applying Young’s inequality at the r.h.s. and optimizing over h1, . . . , hm ∈ Test(M )according to (8.1) yields

∣∣(∇X)sym

∣∣2

HS≤ δ2κ |X |2

2+

⟨X, (~∆X)♯

⟩−

∣∣(∇X)asym

∣∣2

HSm-a.e.

which is the remaining claim (8.3) by the decomposition (1.11).

The following consequence of Lemma 8.2 is not strictly needed in Theorem 8.9,but gives an idea about the reasoning for Lemma 8.8 below.

Corollary 8.3. For every f ∈ D(∆), we have |∇f | ∈ F and

E2(f) ≤ˆ

M

(∆f)2 dm −⟨κ

∣∣ |∇f |2⟩.

Proof. Since D(∆) ⊂ Dreg(Hess) by Corollary 5.12 and ∇Dreg(Hess) ⊂ H1,2(TM )by Theorem 6.3, Lemma 6.13 implies that |∇f | ∈ F whenever f ∈ D(∆). Inparticular, E2(f) is finite and the pairing

⟨κ

∣∣ |∇f |2⟩

is well-defined for every suchf .

The claimed estimate for f ∈ Test(M ) follows by integrating Lemma 8.2 forX := ∇f ∈ Reg(TM ) over all of M , and then using (3.2) and (4.5). In the moregeneral case f ∈ D(∆), let (fk)k∈N be a sequence in Test(M ) as constructed in theproof of Corollary 5.12, i.e. which satisfies E2(fk) → E2(f), ∆fk → ∆f in L2(M )and ∇fk → ∇f in L2(TM ) as k → ∞. The first and the third convergence, withTheorem 6.3, imply that ∇fk → ∇f in H1,2(TM ) as k → ∞. Hence

limk→∞

⟨κ

∣∣ |∇fk|⟩

=⟨κ

∣∣ |∇f |⟩

by Corollary 6.14. The conclusion follows easily.

For Lemma 8.8, but also in Subsection 8.2.2 and Theorem 8.28 below, we shallneed the subsequent Lemma 8.4. (Recall Section 4.1 for the well-definedness of Eqκ

for κ ∈ K1−(M ) and q ∈ [1, 2].) Corollary 8.6 is then deduced along the same linesas Proposition 4.24 after setting q = 1 and letting ε → 0 in Lemma 8.4.

Lemma 8.4. Let X ∈ Reg(TM ), and let φ ∈ D(∆qκ) ∩ L∞(M ) be nonnegativewith ∆qκφ ∈ L∞(M ). Given any ε > 0, we define ϕε ∈ C∞([0,∞)) by ϕε(r) :=(r + ε)q/2 − εq/2. Then for every q ∈ [1, 2],

ˆ

M

[ϕε |X |2

]∆qκφdm

≥ −2ˆ

M

φ[ϕ′ε |X |2

] ⟨X, (~∆X)♯

⟩dm

+ 2⟨κ

∣∣φ |X |2 ϕ′ε |X |2

⟩− q

⟨κ

∣∣φϕε |X |2⟩.

ˆ

M


Proof. According to our choice of q, we have 2r ϕ′′ε (r) ≥ −ϕ′

ε(r) for every r ≥ 0.Recall from Lemma 6.13 and Proposition 1.12 that ϕ′

ε |X |2 ∈ Fb. Therefore, by(4.1), Proposition 1.12, Lemma 8.2 and Lemma 6.13 we get

12

ˆ

M

[ϕε |X |2

]∆qκφdm −

⟨κ

∣∣φ |X |2 ϕ′ε |X |2

⟩

= −12

ˆ

M

[ϕ′ε |X |2

] ⟨∇φ,∇|X |2

⟩dm

−⟨κ

∣∣φ[q(ϕε |X |2)/2 + |X |2 ϕ′

ε |X |2]⟩ˆ

M

= −12E

2κ(φϕ′

ε |X |2, |X |2)

+12

ˆ

M

φ[ϕ′′ε |X |2

] ∣∣∇|X |2∣∣2

dm

− q

2

⟨κ

∣∣φϕε |X |2⟩ˆ

M

≥ˆ

M

φ[ϕ′ε |X |2

] ∣∣∇X∣∣2

HSdm −

ˆ

M

φ[ϕ′ε |X |2

] ⟨X, (~∆X)♯

⟩dm

+ 2ˆ

M

φ[ϕ′′ |X |2

]|X |2

∣∣∇|X |∣∣2

dm − q

2

⟨κ

∣∣φϕε |X |2⟩

≥ −ˆ

M

φ[ϕ′ε |X |2

] ⟨X, (~∆X)♯

⟩dm − q

2

⟨κ

∣∣φϕε |X |2⟩.

Multiplying this inequality by 2 terminates the proof.

Remark 8.5. Let Assumption 8.36 below hold. Then, after partial integration ofthe Hodge Laplacian term and approximation, the conclusion of Lemma 8.4, withκ replaced by κn, n ∈ N, holds under the more general hypothesis q = 2, ε = 0,X ∈ D(~∆)♯ and φ ∈ TestL∞(M ). See also Lemma 8.39 below.

Corollary 8.6. For every X ∈ Reg(TM ),

∆2κ|X |2[M ] = −2⟨κ

∣∣ |X |2⟩.

We will have to identify 1-forms in H1,2(T ∗M ) with their vector field coun-terparts in while additionally retaining their respective first order regularities inTheorem 8.9. This is discussed now. In some sense, Lemma 8.8 can be seen as ananalogue of Gaffney’s inequality in Remark 7.20 under curvature lower bounds.

Definition 8.7. We define the space H1,2♯ (TM ) as the image of H1,2(T ∗M ) under

the map ♯, endowed with the norm

‖X‖H1,2♯

(TM) := ‖X‖H1,2(T∗M).

Lemma 8.8. H1,2♯ (TM ) is a subspace of H1,2(TM ). The aforementioned natural

inclusion is continuous. Additionally, for every X ∈ H1,2♯ (TM ),

Ecov(X) ≤ Econ(X) −⟨κ

∣∣ |X |2⟩.

Proof. Let ρ′ ∈ [0, 1) and α′ ∈ R be as in Lemma 4.5 for µ := κ−.Clearly Reg(TM ) = Reg(T ∗M )♯ ⊂ H1,2

♯ (TM ) as well as Reg(TM ) ⊂ H1,2(TM )by definition of the respective spaces. Moreover, the claimed inequality for X ∈Reg(TM ) follows after evaluating the inequality in Lemma 8.2 at M and usingLemma 8.1 and Corollary 8.6.

To prove that H1,2♯ (T ∗M ) ⊂ H1,2(TM ) with continuous inclusion, we again first

study what happens for X ∈ Reg(TM ). By the previous step, Lemma 4.5 forµ := κ− together with Lemma 6.13, and (7.3),

Ecov(X) ≤ −⟨κ

∣∣ |X |2⟩

+Econ(X) (8.4)

98 MATHIAS BRAUN

≤⟨κ−

∣∣ |X |2⟩

+Econ(X)

≤ ρ′Ecov(X) + α′∥∥X

∥∥2

L2(TM)+Econ(X).

Rearranging yields

Ecov(X) ≤ α′

1 − ρ′

∥∥X∥∥2

L2(TM)+

11 − ρ′

Econ(X). (8.5)

This inequality extends to arbitrary X ∈ H1,2♯ (T ∗M ) by applying it to all members

of a sequence (Xn)n∈N in Reg(TM ) such that Xn → X in H1,2♯ (TM ) as n → ∞

and using the L2-lower semicontinuity of Ecov from Theorem 6.3. In particular, asthe r.h.s. of (8.5) is continuous in H1,2

♯ (TM ), both H1,2♯ (TM ) ⊂ H1,2(TM ) and the

continuity of this inclusion follow. In particular, by Corollary 6.14 the inequality(8.4) is stable under this limit procedure.

8.1.2. Main result. We now come to the main result of this article.

Theorem 8.9. There exists a unique continuous mapping Ricκ : H1,2♯ (TM )2 →

M±f (M )E satisfying the identity

Ricκ(X,Y ) = ∆2κ 〈X,Y 〉2

+[1

2

⟨X, (~∆Y )♯

⟩

+12

⟨Y, (~∆X)♯

⟩− ∇X : ∇Y

]m

(8.6)

for every X,Y ∈ Reg(TM ). The map Ricκ is symmetric and R-bilinear. Further-more, for every X,Y ∈ H1,2

♯ (TM ), it obeys

Ricκ(X,X) ≥ 0,

Ricκ(X,Y )[M ] =ˆ

M

[〈dX, dY 〉 + δX δY

]dm

−ˆ

M

∇X : ∇Y dm −⟨κ

∣∣ 〈X,Y 〉⟩,

∥∥Ricκ(X,Y )∥∥2

TV≤

[Econ(X) +

⟨κ

∣∣ |X |2⟩] [

Econ(Y ) +⟨κ

∣∣ |Y |2⟩].

(8.7)

Proof. Given any X,Y ∈ Reg(TM ), we define Ricκ(X,Y ) ∈ M±f (M )E by (8.6).

This assignment is well-defined since 〈X,Y 〉 ∈ D(∆2κ) by Lemma 8.2 and gives anonnegative element. The map Ricκ : Reg(TM )2 → M

±f (M )E defined in that way

is clearly symmetric and R-bilinear.Let X and Y as above. Then by Lemma 8.1 and Corollary 8.6,

‖Ricκ(X,X)‖TV = Ricκ(X,X)[M ]

= Econ(X) −Ecov(X) −⟨κ

∣∣ |X |2⟩

≤ Econ(X) −⟨κ

∣∣ |X |2⟩.

By symmetry and R-bilinearity of Ricκ, for every λ ∈ R we obtain

λ2 Ricκ(X,X) + 2λRicκ(X,Y ) + Ricκ(Y, Y )

= Ricκ(λX + Y, λX + Y ) ≥ 0.

Lemma 5.8 thus implies that

‖Ricκ(X,Y )‖TV ≤[Econ(X) −

⟨κ

∣∣ |X |2⟩]1/2 [

Econ(Y ) −⟨κ

∣∣ |Y |2⟩]1/2

.

Since the r.h.s. is jointly continuous in X and Y w.r.t. convergence in H1,2♯ (TM )

by Lemma 8.8 and Corollary 6.14, the above map Ricκ extends continuously anduniquely to a (non-relabeled) map Ricκ : H1,2

♯ (TM )2 → M±f (M ).


In particular, the last inequality of (8.7) directly comes as a byproduct of theprevious argument, while the first inequality of (8.7) follows by continuously ex-tending Lemma 8.2 to any X ∈ H1,2

♯ (TM ). The second identity for X ∈ Reg(TM )follows from Lemma 8.1 and Corollary 8.6 after integration by parts. Since bothsides are jointly H1,2

♯ -continuous in X and Y by Lemma 8.8 and Corollary 6.14,this identity easily extends to all X,Y ∈ H1,2

♯ (TM ).

Remark 8.10. In the abstract setting of diffusion operators, Sturm [96] introduced a“pointwise”, possibly dimension-dependent Ricci tensor. Although the smoothnessassumptions are not really justified in our setting [45, Rem. 3.6.6], it would beinteresting to study the behavior of Ricκ under conformal and drift transformationsof 〈·, ·〉 and m, respectively, as done in the abstract framework of [96].

8.2. Curvature tensors from the κ-Ricci measure. Next, we separate κ fromRicκ in Theorem 8.9 to define the Ricci curvature and the second fundamental formintrinsically defined by the data (M ,E,m).

8.2.1. Measure-valued Laplacian.

Definition 8.11. We define D(∆) to consist of all f ∈ F for which ∇f ∈DL2 (div), in which case we define the measure-valued Laplacian of f by

∆f := div ∇f.Lemma 8.12. The signed Borel measure u2 κ has finite total variation for everyu ∈ F. In particular, if u2 ∈ D(∆2κ), we have u2 ∈ D(∆) with

∆(u2) = ∆2κ(u2) + 2 u2 κ.

In particular ∆|∇f |2 has finite total variation for every f ∈ Test(M ).

Proof. The first claim follows by observing that κ ∈ K1−(M ) if and only if |κ| ∈K1−(M ), whence u2 |κ| is a finite measure by [41, Cor. 2.25].

The remaining claims follow by direct computations using Definition 4.19, (4.1)and Proposition 4.24.

8.2.2. Dimension-free Ricci curvature. Keeping in mind Theorem 8.9, we define themap Ric : H1,2

♯ (TM )2 → M±f (M )E by

Ric(X,Y ) := Ricκ(X,Y ) + 〈X,Y 〉∼ κ.

This map is well-defined, symmetric and R-bilinear — indeed, by polarization,Lemma 8.8 and Lemma 6.13, 〈X,Y 〉∼ κ ∈ M

±f (M )E for every X,Y ∈ H1,2

♯ (TM ).Ric is also jointly continuous, which follows from polarization, Corollary 6.14 andLemma 8.8 again. Lastly, by Theorem 8.9, Lemma 8.2 and Lemma 8.12, for everyX,Y ∈ Reg(TM ),

Ric(X,Y ) = ∆〈X,Y 〉

2+

[12

⟨X, (~∆Y )♯

⟩

+12

⟨X, (~∆Y )♯

⟩− ∇X : ∇Y

]m.

(8.8)

Of course, (8.8) is defined to recover the familiar vector Bochner formula

∆|X |2

2+

⟨X, (~∆X)♯

⟩m = Ric(X,X) +

∣∣∇X∣∣2

HSm

for X ∈ Reg(TM ), which, setting X := ∇f , f ∈ Test(M ) and using Lemma 8.1 aswell as Theorem 6.3, in turn reduces to the Bochner identity

∆|∇f |2

2−

⟨∇f,∇∆f

⟩m = Ric(∇f,∇f) +

∣∣Hess f∣∣2

HSm.

100 MATHIAS BRAUN

In general, Ric(X,X) is no longer nonnegative, but it is bounded from below byκ in the following sense, which is a direct consequence of (8.8), (8.7) as well as therespective H1,2

♯ -continuity of both sides of the resulting inequality.

Proposition 8.13. For every X ∈ H1,2♯ (TM ), the map Ric defined above obeys

Ric(X,X) ≥ |X |2∼ κ.It is unclear whether (8.8) or (8.6) hold for generalX,Y ∈ H1,2

♯ (TM ), since we donot know whether 〈X,Y 〉 ∈ D(∆) or 〈X,Y 〉 ∈ D(∆2κ) in the respective situation.Still, (8.8) makes sense in the subsequent weak form. See also Remark 8.27 below.

Lemma 8.14. For every X,Y ∈ H1,2♯ (TM ) and every f ∈ Test(M ),

ˆ

M

f dRic(X,Y ) =ˆ

M

[⟨dX, d(f Y )

⟩+ δX δ(f Y ) − ∇X : ∇(f Y )

]dm

=ˆ

M

[Hess f(X,Y ) + df(X divY + Y divX)

]dm

+ˆ

M

f[⟨

dX, dY ⟩

+ δX δY − ∇X : ∇Y]

dm.

Proof. For a given f ∈ Test(M ), all terms are continuous in X and Y w.r.t. con-vergence in H1,2

♯ (TM ). Hence, without restriction, we may and will assume in thesequel that X,Y ∈ Reg(TM ).

Under these assumptions, we first claim thatˆ

M

dX(∇f, Y ) dm

=ˆ

M

[−〈X,Y 〉 ∆f + 〈X,∇f〉 divY −

⟨X, [∇f, Y ]

⟩]dm.

(8.9)

If Y ∈ Test(TM ) and f is the product of two elements of Test(M ), by Proposition 1.12,we have ∇f ∈ Test(TM ), and hence (8.9) simply follows from the Definition 7.2 ofthe exterior derivative d. To relax the assumption on Y , note that since 〈X,∇f〉 ∈F by Lemma 6.13, by definition of the Lie bracket, Theorem 6.3, Proposition 6.11and (3.2),

ˆ

M

dX(∇f, Y ) dm

=ˆ

M

[−〈X,Y 〉 ∆f − d〈X,∇f〉(Y ) − ∇Y : (∇f ⊗X)

]dm

+ˆ

M

Hess f(Y,X) dm

=ˆ

M

[−〈X,Y 〉 ∆f − d〈X,∇f〉(Y ) − d〈Y,X〉(∇f)

]dm

+ˆ

M

[∇X : (∇f ⊗ Y ) + Hess f(Y,X)

]dm

=ˆ

M

[−d〈X,∇f〉(Y ) + ∇X : (∇f ⊗ Y ) + Hess f(Y,X)

]dm.

Both extremal sides of this identity are L2-continuous in Y , whence it extends toany Y ∈ Reg(TM ). For such Y , the second and the third identity still hold —compare with the above mentioned results — and (8.9) follows since Y ∈ D(div)by (3.2), (3.3) and Lemma 3.15, whence by Definition 3.5,

−ˆ

M

d〈X,∇f〉(Y ) dm =ˆ

M

〈X,∇f〉 div Y dm.


Similarly, the assumption on f is relaxed now. Indeed, by (8.9),ˆ

M

dX(∇f, Y ) dm

=ˆ

M

[d〈X,Y 〉(∇f) − 〈X,∇f〉 divY − ∇Y : (∇f ⊗X)

]dm

+ˆ

M

Hess f(Y,X) dm

=ˆ

M

[d〈X,Y 〉(∇f) − 〈X,∇f〉 divY − ∇Y : (∇f ⊗X)

]dm

+ˆ

M

[d〈∇f,X〉(Y ) − ∇X : (Y ⊗ ∇f)

]dm

=ˆ

M

[d〈X,Y 〉(∇f) − 〈X,∇f〉 divY − ∇Y : (∇f ⊗X)

]dm

−ˆ

M

[〈∇f,X〉 div Y + ∇X : (Y ⊗ ∇f)

]dm.

Here, we successively used that 〈X,Y 〉, 〈∇f,X〉 ∈ Fby Lemma 6.13, the definitionof the Lie bracket, Theorem 6.3, Proposition 6.11, and that Y ∈ D(div) by (3.2),(3.3) and Lemma 3.15. Using Lemma 4.17, both extremal sides of this identity thusextend to general f ∈ Test(M ). As above, the second and the third equality stillhold under this assumption. Thus, (8.9) is proven in the desired generality.

We turn to the claim of the lemma and initially prove the second equality. First,using Theorem 7.5, (8.9), the definition of the Lie bracket and Theorem 6.3,

ˆ

M

⟨dX, d(f Y )

⟩dm

=ˆ

M

[dX(∇f, Y ) + f

⟨dX, dY

⟩]dm

=ˆ

M

[−〈X,Y 〉 ∆f + 〈X,∇f〉 divY −

⟨X, [∇f, Y ]

⟩]dm

+ˆ

M

f⟨dX, dY

⟩dm

=ˆ

M

[−〈X,Y 〉 ∆f + 〈X,∇f〉 divY − ∇Y : (∇f ⊗X)

]dm

+ˆ

M

[Hess f(X,Y ) + f

⟨dX, dY

⟩]dm.

Second, by (7.3), Lemma 3.13 and Lemma 3.15,ˆ

M

δX δ(f Y ) dm =ˆ

M

divX div(f Y ) dm

=ˆ

M

[divX 〈∇f, Y 〉 + f divX div Y

]dm.

Third, by Lemma 6.9,

−ˆ

M

∇X : ∇(f Y ) dm = −ˆ

M

[∇X : (∇f ⊗ Y ) − f ∇X : ∇Y

]dm.

Adding up these three identities and employing thatˆ

M

[−〈X,Y 〉 ∆f − ∇Y : (∇f ⊗X) − ∇X : (∇f ⊗ Y )

]dm = 0

thanks to (3.2), Lemma 6.13 and Proposition 6.11, the second claimed equality inthe lemma is shown.

102 MATHIAS BRAUN

To prove the first identity, denote the r.h.s. of it by A(X,Y ). By what we haveproved above, it follows that A(X,Y ) = A(Y,X). Since f X, f Y ∈ H1,2

♯ (TM ) byTheorem 7.5, by Lemma 8.12 and (8.8) we get

ˆ

M

f dRic(X,Y ) = −12

ˆ

M

[⟨∇f,∇〈X,Y 〉

⟩+ 2f ∇X : ∇Y

]dm

+12

ˆ

M

[⟨d(f X), dY

⟩+ δ(f X) δY

]dm

+12

ˆ

M

[⟨dX, d(f Y )

⟩+ δX δ(f Y )

]dm

= −12

ˆ

M

[∇(f X) : ∇Y + ∇X : ∇(f Y )

]dm

+12

ˆ

M

[⟨d(f X), dY

⟩+ δ(f X) δY

]dm

+12

ˆ

M

[⟨dX, d(f Y )

⟩+ δX δ(f Y )

]dm

=12

[A(X,Y ) + A(Y,X)

]= A(X,Y ).

In the second step, we used Proposition 6.11 and then Lemma 6.9.

Lemma 8.15. For every X,Y ∈ H1,2♯ (TM ) and every f ∈ Test(M ),

Ric(f X, Y ) = f Ric(X,Y ).

Proof. By the H1,2♯ -continuity of both sides in X and Y (recall Theorem 7.5 and

Lemma 7.15), we may and will assume without restriction that X,Y ∈ Reg(TM ).Owing to Lemma 4.18, it moreover suffices to prove that for every g ∈ Test(M ),

ˆ

M

g dRic(f X, Y ) =ˆ

M

g f dRic(X,Y ). (8.10)

As in the proof of Lemma 8.14 above, by (8.8) and Lemma 8.1 we haveˆ

M

g dRic(f X, Y ) =12

ˆ

M

[∆g f 〈X,Y 〉 + g f

⟨X, (~∆Y )♯

⟩]dm

+12

ˆ

M

[g

⟨Y, (~∆(f Y ))♯

⟩− 2g∇(f X) : ∇Y

]dm,

ˆ

M

g f dRic(X,Y ) =12

ˆ

M

[∆(g f) 〈X,Y 〉 + g f

⟨X, (~∆Y )♯

⟩]dm

+12

ˆ

M

[g f

⟨Y, (~∆X)♯

⟩− 2g f ∇X : ∇Y

]dm.

Using Lemma 1.16 and Lemma 6.9 yields

∆(g f) = g∆f + 2 〈∇g,∇f〉 + f ∆g m-a.e.,

∇(f X) : ∇Y = f ∇X : ∇Y + (∇f ⊗X) : ∇Y m-a.e.,

while Lemma 8.1 ensures that

(~∆(f X))♯ = f (~∆X)♯ − ∆f X − 2 ∇∇fX.

Lastly, since 〈X,Y 〉 ∈ Fb by Lemma 6.13, 〈X,Y 〉 ∇f ∈ DTV(div) ∩ D(div) withn(〈X,Y 〉 ∇f) = 0 by Lemma 3.15. Together with Proposition 6.11, this yields

ˆ

M

〈∇g,∇f〉〈X,Y 〉 dm = −ˆ

M

g div(〈X,Y 〉 ∇f

)dm

= −ˆ

M

[g∆f 〈X,Y 〉 + ∇X : (∇f ⊗ Y )

]dm


+ˆ

M

∇Y : (∇f ⊗X) dm.

Using the last four identities, a term-by-term comparison in the above identities forboth sides of (8.10) precisely yield (8.10).

Following Subsection 1.1.1, denote by Ric≪ : H1,2♯ (TM )2 → M

±f (M )E the con-

tinuous map — recall (1.1) — which assigns to Ric(X,Y ) its m-absolutely contin-uous part Ric≪(X,Y ) := Ric(X,Y )≪, X,Y ∈ H1,2

♯ (TM ).

Definition 8.16. Either of the maps Ric≪ or ric : H1,2♯ (TM )2 → L1(M ) given by

ric(X,Y ) :=dRic≪(X,Y )

dm

is called Ricci curvature of (M ,E,m).

The advantage of this definition is threefold. First, the map ric extends to all ofL2(TM )2 in a sense made precise in Theorem 8.21 below in the finite-dimensionalframework. Second, this separation of Ric into m-absolutely continuous and m-singular parts allows us to define the second fundamental form from the latter inDefinition 8.23 below. Third, it makes it possible to say that the negative part ofthe “lowest eigenvalue” of ric satisfies the extended Kato condition according toProposition 8.17. To this aim, define ric∗ : M → [−∞,∞] by

ric∗ := essinf

|X |−2 ric(X,X) : X ∈ H1,2♯ (TM ), |X | ≤ 1 m-a.e.

.

Proposition 8.17. The measure ric−∗ m belongs to K1−(M ).

Proof. By [41, Rem. 2.8], ric−∗ m ∈ K1−(M ) if and only if the function ric−

∗ belongsto the extended functional Kato class of M [41, Def. 2.20]. The proof of the latterwill in particular yield ric−

∗ ∈ L1loc(M ) and ric−

∗ m ∈ M+σ (M )E as a byproduct.

Let κ = κ+ −κ− be the Jordan decomposition of κ with κ+, κ− ∈ M+σ (M ). Since

a|κ|t ≥ aκ

−

t for every t ≥ 0, we have κ− ∈ K1−(M ) and dκ−/dmm ∈ K1−(M ) by(1.1). Hence, to prove the claim it is sufficient to prove that ric−

∗ ≤ dκ−/dm m-a.e.Indeed, given any X ∈ H1,2

♯ (TM ), by definition of ric(X,X) and Theorem 8.9,

ric(X,X) ≥ |X |2 dκdm

≥ −|X |2 dκ−

dmm-a.e.

In particular ric−∗ < ∞ m-a.e., and thus

ric−∗ ≤ esssup

|X |−2 ric(X,X)− : X ∈ H1,2

♯ (TM ), |X | ≤ 1 m-a.e.

≤ dκ−

dmm-a.e.

8.2.3. Dimension-dependent Ricci tensor. Following the RCD∗(K,N)-treatise [53],K ∈ R, and motivated by [9], we shortly outline the definition of an N -Ricci tensoron BE2(κ,N) spaces for N ∈ [1,∞). The latter condition is assumed to hold for(M ,E,m) throughout this subsection. Details are left to the reader.

Keeping in mind Proposition 5.14, let (En)n∈N∪∞ be the dimensional decom-position of L2(TM ) and define the function dimloc : M → 1, . . . , ⌊N⌋ by

dimloc := 1E1 + 2 1E2 + · · · + ⌊N⌋ 1⌊N⌋.

Arguing as for [53, Prop. 4.1], we see that for every X ∈ H1,2♯ (TM ), if N ∈ N then

tr ∇X = divX m-a.e. on EN

104 MATHIAS BRAUN

according to the definition (1.14) of the trace of a generic A ∈ L2(T⊗2M ). Thefunction RN : H1,2

♯ (TM )2 → R defined by

RN (X,Y ) :=

[tr ∇X − divX

] [tr ∇Y − divY

]

N − dimlocif dimloc < N,

0 otherwise

is thus well-defined. In fact, the assignment (X,Y ) 7→ RN (X,Y )m is continuousas a map from H1,2

♯ (TM )2 into M±f (M )E thanks to Lemma 8.8.

Definition 8.18. The map RicN : H1,2♯ (TM )2 → M

±f (M )E given by

RicN (X,Y ) := Ric(X,Y ) − RN (X,Y )m

is henceforth called N -Ricci tensor of (M ,E,m).

Theorem 8.19. For every X ∈ H1,2♯ (TM ),

RicN (X,X) ≥ |X |2∼ κ,

∆|X |2

2+

⟨X, (~∆X)♯

⟩≥ RicN (X,X) +

1N

|divX |2 m.

Proof. We only outline the main differences to the proof of [53, Thm. 4.3]. Set

RicκN (X,Y ) := Ricκ(X,Y ) − RN (X,Y )m

for X,Y ∈ H1,2♯ (TM ). By Lemma 8.2 and absolute continuity of RN (X,Y ) w.r.t. m,

we see that RicκN (X,X)⊥ ≥ 0, understood w.r.t. m, for every such X . The non-negativity of RicκN (X,X)≪ for X := ∇f , f ∈ Test(M ), is argued similarly to [53,Thm. 3.3] up to replacing Γ2(f) therein by Γ2κ

2 (f), compare with Lemma 5.9. ByLemma 8.15 and the definition of Ric, it follows that Ricκ is both R- and Test-bilinear. The same is true for RN by Lemma 6.9 and the definition (1.14) of thetrace. Proceeding now as in the proof of [53, Thm. 4.3] implies the nonnegativityof RicκN (X,X) for every X ∈ Reg(TM ), and hence for every X ∈ H1,2

♯ (TM ) bycontinuity. We conclude the first inequality from the definition of Ric again. Theargument for the second is the same as in [53].

We finally turn to the existence proof of an appropriate extension of ric toL2(TM )2 announced after Definition 8.16. By Test-bilinearity of RicN encoun-tered in the above proof of Theorem 8.19, a similar statement holds for the mapinduced by d(RicN )≪/dm but is not written down for notational convenience.

The quite technical proof of the following preparatory result is the same as for[47, Lem. A.1, Thm. A.2] — up to replacing the norm ‖ · ‖W 1,2(TM) by ‖ · ‖H1,2

♯(TM)

— and thus omitted.

Lemma 8.20. Denote by (En)n∈N∪∞ the dimensional decomposition of L2(TM )according to Proposition 1.29. Then there exist V1, . . . , V⌊N⌋ ∈ H1,2

♯ (TM ) such thatV1, . . . , Vn is a local basis of L2(TM ) on En for every n ∈ 1, . . . , ⌊N⌋.

Theorem 8.21. There exists a unique symmetric and L∞-bilinear assignmentric : L2(TM )2 → L1

loc(M ) whose restriction to H1,2♯ (TM )2 coincides with ric.

Proof. We first define ric(f X, Y ) for given X,Y ∈ H1,2♯ (TM ) and f ∈ L∞(M ). Let

(fn)n∈N be a sequence in Test(M ) such that fn → f pointwise m-a.e. as n → ∞ aswell as supn∈N

‖fn‖L∞ < ∞. Sinceˆ

M

∣∣ric(fnX,Y ) − ric(fmX,Y )∣∣ dm =

ˆ

M

|fn − fm|∣∣ric(X,Y )

∣∣ dm

by Lemma 8.15 for every n,m ∈ N, (ric(fnX,Y ))n∈N is a Cauchy sequence inL1(M ) and hence converges to a limit denoted by ric(f X, Y ) ∈ L1(M ). By an


analogous argument, we see that this procedure does not dependent on the cho-sen sequence in Test(M ) with the above properties, and moreover ric(f X, Y ) =ric(f X, Y ) if f ∈ Test(M ). The symmetry and bilinearity of ric thus straightfor-wardly induces a symmetric and L∞-bilinear map ric : W2 → L1(M ). Here, W⊂L2(TM ) is the linear span of all elements of the form f X , f ∈ L∞(M ) andX ∈ H1,2

♯ (TM ).Now we define ric(X,Y ) for general X,Y ∈ L2(TM ). Retaining the notation of

Lemma 8.20, let n ∈ N, and let C ∈ B(M ) be some subset of En such that forsome f1, . . . , fn, g1, . . . , gn ∈ L∞(M ) vanishing m-a.e. outside C,

1C X = f1 V1 + · · · + fn Vn,

1C Y = g1 V1 + · · · + gn Vn.(8.11)

Locally, we then define

1C ric(X,Y ) :=n∑

i,j=1

fi gj ric(Vi, Vj).

This is a good definition that does not depend on the representations (8.11) of Xand Y on C and D, respectively, and is easily seen to give rise to a (non-relabeled)map ric : L2(TM )2 → L1

loc(M ) which has all desired properties.

Remark 8.22. By the local definition of ric in Theorem 8.21, we neither know if theintegrability of ric(X,Y ) for given X,Y ∈ L2(TM ) can be improved, nor if ric isactually continuous in an appropriate target topology.

8.2.4. Second fundamental form. Similarly as for Definition 8.16, let us denote byRic⊥ : H1,2

♯ (TM )2 → M±f (M )E the continuous map — recall (1.1) — assigning to

Ric(X,Y ) its m-singular part Ric⊥(X,Y ) := Ric(X,Y )⊥, X,Y ∈ H1,2♯ (TM ).

Definition 8.23. The map II : H1,2♯ (TM )2 → M

±f (M )E given by

II(X,Y ) := Ric⊥(X,Y )

is called second fundamental form of (M ,E,m).

In particular, if X ∈ Reg(TM ), then by (8.8), Definition 8.11 and Definition 3.8,

II(X,X) = ∆⊥|X |2

2= div⊥ ∇|X |2

2= − n ∇|X |2

2. (8.12)

This observation complements the nonsmooth “boundary” discussion pursued so far.In particular, (8.12) is fully justified in the smooth context, as noted in the nextexample (and it only depends on the “tangential parts” of X and Y by definitionof H1,2

♯ (TM ), recall Lemma 3.15). See [55, Ch. 2] for similar computations.

Example 8.24. Let M be a Riemannian manifold with boundary ∂M . The secondfundamental form of ∂M is the map I defined by

I(X‖, Y ‖) := 〈∇X‖n, Y ‖〉for X‖, Y ‖ ∈ Γ(T∂M ). According to Section 1.2, such an X‖ can be uniquely iden-tified with X ∈ Γ(TM )

∣∣∂M

such that 〈X, n〉 = 0 on ∂M . We extend X and n to(non-relabeled) smooth vector fields defined on an open neighborhood of ∂M [74,Lem. 8.6]. Then by metric compatibility of the Levi-Civita connection on M ,

I(X‖, X‖) = −〈∇XX, n〉 +⟨∇X,∇〈X, n〉

⟩= −1

2

⟨∇|X |2, n

⟩on ∂M .

With Example 3.9, this shows that for every smooth, compactly supported andpurely tangential X ∈ Γ(TM ) to which X‖ ∈ Γ(T∂M ) is uniquely associated,

II(X,X) = I(X‖, X‖) s.

106 MATHIAS BRAUN

In line with Example 8.24, we make the following bibliographical remark which,in fact, partly motivated Definition 8.23.

Remark 8.25. Definition 8.16 and Definition 8.23 together yield the identity

Ric(X,X) = ric(X,X)m + II(X,X)

for every X ∈ H1,2♯ (TM ). This identity — and hence our definitions — have a

smooth evidence by [55, Thm. 2.4]. There it has been shown that on any smooth,connected Riemannian manifold M with boundary (with measure m := e−2w

v, w ∈C2(M ), so that s = e−2wH

d−1∣∣∂M

), with Ric the Ricci measure in the sense of [45,Thm. 3.6.7] — and hence of Subsection 8.2.2 — we have

Ric(∇f,∇f) = Ric(∇f,∇f)m + I(∇f‖,∇f‖) s

for every f ∈ C∞c (M ) with df(n) = 0 on ∂M .

Remark 8.26 (Convexity of RCD spaces). In the novel interpretation proposed byDefinition 8.23, on an RCD(K,∞) space, K ∈ R, [45, Thm. 3.6.7] implies thatevery such space is intrinsically convex in the sense that

II(X,X) ≥ 0 (8.13)

for every X ∈ H1,2♯ (TM ). In other words, every such space is necessarily convex in

the sense that (8.13) holds for, say, every X ∈ Reg(TM ). (This notion of convexityis frequently used the smooth setting, see e.g. [102, Def. 1.2.2].)

This should not be surprising from various perspectives.First, we know from [5, Thm. 6.18] that geodesic convexity of a subset Y of

M is a sufficient condition for it to naturally become again RCD(K,∞) as soonas m[∂Y ] = 0 and m[Y ] > 0. Through (8.13) and [55] we have thus provideda nonsmooth analogue of the fact that every, say, compact, geodesically convexRiemannian manifold with boundary has nonnegative second fundamental form[84, Lem. 61]. (The converse, of course, does not hold in general, e.g. for disks onthe cylinder S1 × R with diameter larger than π.)

Second, recent results [98] and examples [102] show that on nonconvex domains— even if the boundary has arbitrarily small concavity — show that in general, onecannot expect uniform lower Ricci bounds solely described by relative entropies.

The second fundamental form really matters when one is tempted to derive aWeitzenböck formula from Lemma 8.14. In other words, the latter should really betreated as an interior identity, away from singularities [84, Cor. 21].

Remark 8.27 (Weitzenböck identity). If X ∈ D(~∆)♯ ∩ D() in Lemma 8.14, fromthe latter we could deduce that

ˆ

M

f dRic(X,Y ) =ˆ

M

f⟨Y, (~∆X)♯ + X

⟩dm

for every Y ∈ H1,2♯ (TM ), which is strongly reminiscent of the Weitzenböck formula

[84, Cor. 21]. In other words, Ric(X,Y ) ≪ m and

ric(X,Y ) =⟨Y, (~∆X)♯ + X

⟩m-a.e., (8.14)

which, if Y ∈ D(~∆)♯ ∩ D() as well, especially implies the pointwise symmetry⟨Y, (~∆X)♯ + X

⟩=

⟨(~∆Y )♯ + Y,X

⟩m-a.e.

The identity (8.14) plays a crucial role in deriving the Feynman–Kac formula forthe semigroup on differential 1-forms on Riemannian manifolds, with or withoutboundary, as well as Bismut–Elworthy–Li formulas for dptf , f ∈ L2(M ) ∩ L∞(M )and t > 0 [11, 38, 67, 102]. Still, we do not know whether D(~∆)♯ ∩ D() 6= 0.


Let us remark that for X ∈ H1,2♯ (TM ) to belong to both D(~∆)♯ and D(), from

(8.14) one would necessarily have II(X, ·) = 0. On a compact Riemannian manifoldM with boundary, this is underlined by comparison of the boundary conditions for~∆ and . Indeed, let X ∈ Γ(TM ). By (6.7), recall that X ∈ D() means that

X⊥ = 0,

(∇nX)‖ = 0

on ∂M according to (1.2). On the other hand, X ∈ D(~∆) entails absolute boundaryconditions as in Remark 7.19 for X which, by [67, Lem. 4.1], are equivalent to

X⊥ = 0,

(∇nX)‖ − I(X‖, ·) = 0

at ∂M . Hence, we must have I(X‖, ·) = 0 on ∂M .

8.3. Vector Bochner inequality. The subsequent vector q-Bochner inequality isa direct consequence from Lemma 8.4, q ∈ [1, 2]. For RCD(K,∞) or RCD∗(K,N)spaces, K ∈ R and N ∈ [1,∞), it is due to [16, Thm. 3.13] for q = 1. Note thatthe assumption of Theorem 8.28 is satisfied if X ∈ Test(TM ) by Lemma 8.1.

Let D(∆qκ) be defined w.r.t. the closed form Eqκ as in Definition 4.19, q ∈ [1, 2].

Theorem 8.28. Suppose that X ∈ Reg(TM ) satisfies ~∆X ∈ L1(T ∗M ). Then forevery q ∈ [1, 2], we have |X |q ∈ D(∆qκ) and

∆qκ |X |qq

+ |X |q−2⟨X, (~∆X)♯

⟩m ≥ 0.

Proof. Letting ε → 0 in Lemma 8.4 with Lebesgue’s theorem yieldsˆ

M

|X |qq

∆qκφdm ≥ −ˆ

M

φ |X |q−2⟨X, (~∆X)♯

⟩dm

for every φ ∈ D(∆qκ) ∩ L∞(M ) with ∆qκφ ∈ L∞(M ). Since the function on ther.h.s. which involves X belongs to L1(M )∩L2(M ) — and here is where we use that~∆X ∈ L1(M ) — a variant of [41, Lem. 6.2] for E

qκ in place of E2κ implies that|X |q ∈ D(∆qκ) with the desired inequality.

Remark 8.29. If κ ∈ K0(M ), the quadratic form Eqκ is well-defined and closed even

for q ∈ [2,∞), and Theorem 8.28 can be deduced along the same lines for this rangeof q even without the assumption that ~∆X ∈ L1(T ∗M ). Compare e.g. with thefunctional treatise [17, Ch. 3].

8.4. Heat flow on 1-forms. A slightly more restrictive variant of Theorem 8.28yields functional inequalities for the heat flow (ht)t≥0 on 1-forms, see Theorem 8.40.The latter is shortly introduced before, along with its basic properties. A thoroughstudy of (ht)t≥0 on RCD(K,∞) spaces, K ∈ R, has been pursued in [16].

8.4.1. Heat flow and its elementary properties. Analogously to Subsection 1.3.7 andSubsection 6.3.2, we define the heat flow on 1-forms as the semigroup (ht)t≥0 ofbounded, linear and self-adjoint operators on L2(T ∗M ) by

ht := e−~∆t. (8.15)

It is associated [43, Thm. 1.3.1] to the functional Econ : L2(T ∗M ) → [0,∞] with

Econ(ω) :=

ˆ

M

[|dω|2 + |δω|2

]dm if ω ∈ H1,2(T ∗M ),

∞ otherwise.(8.16)

108 MATHIAS BRAUN

Theorem 8.30. The subsequent properties of (ht)t≥0 hold for every ω ∈ L2(T ∗M )and every t > 0.

(i) The curve t 7→ htω belongs to C1((0,∞); L2(T ∗M )) with

ddt

htω = −~∆htω.

(ii) If ω ∈ D(~∆), we have

ddt

htω = −ht~∆ω.

In particular, we have the identity

~∆ ht = ht ~∆ on D(~∆).

(iii) For every s ∈ [0, t],

‖htω‖L2(T∗M) ≤ ‖hsω‖L2(T∗M).

(iv) The function t 7→ Econ(htω) belongs to C1((0,∞)), is nonincreasing, andits derivative satisfies

ddtEcon(htω) = −2

ˆ

M

∣∣~∆htω∣∣2

dm.

(v) If ω ∈ H1,2(T ∗M ), the map t 7→ htω is continuous on [0,∞) w.r.t. strongconvergence in H1,2(T ∗M ).

(vi) We have

Econ(htω) ≤ 12t

∥∥ω∥∥2

L2(TM),

∥∥~∆htω∥∥2

L2(T∗M)≤ 1

2t2∥∥ω

∥∥2

L2(T∗M).

Via the closedness of d from Lemma 2.10 together with Lemma 8.1, the followingLemma 8.31 is verified. The spectral bottom inequality from Corollary 8.35 followsfrom the second identity of (8.7), Lemma 6.13, (4.1) and Rayleigh’s theorem.

Lemma 8.31. For every f ∈ F and every t > 0, dptf ∈ D(~∆) and

htdf = dptf.

Remark 8.32. It is part of the statement of Lemma 8.31 that dptf ∈ H1,2(T ∗M ).Indeed, if f ∈ Fb, we even have dptf ∈ Reg(T ∗M ). Using that by Theorem 7.5 andLemma 7.15, d(dptf) = 0 and δ(dptf) = −∆ptf for such f , the claim for generalelements of F easily follows by truncation and Lemma 1.17.

Remark 8.33. Analogously to (8.15), it is possible to define the heat flow (hkt )t≥0

in L2(ΛkT ∗M ) with generator −~∆k for any k ∈ N. However, it is not clear if thecommutation relation from Lemma 8.31 holds between (hkt )t≥0 and (hk−1

t )t≥0 fork ≥ 2. Compare with [16, Rem. 3.4].

Corollary 8.34. If ω ∈ D(δ) and t > 0, then htω ∈ D(δ) with

δhtω = ptδω.

Corollary 8.35. We have

inf σ(−∆κ) ≤ inf σ(~∆).


8.4.2. Functional inequalities and Lp-properties. Unlike the results for (ht)t≥0 from[16] for RCD(K,∞) spaces, K ∈ R, a collateral effect of the singular potential κis that we do not know how the domains D(∆) and D(∆2κ) are related. Comparewith Remark 8.41 below. We thus restrict ourselves to the following assumptionthroughout this subsection.

Assumption 8.36. In the framework of Assumption 4.10, there exists k ∈ L1loc(M )

in the functional extended Kato class of M [41, Def. 2.20] which is uniformly boundedfrom below by some K ∈ R, such that

κ = km.

We define the sequence (κn)n∈N in K1−(M ) by

κn := knm

with kn := minn,k ∈ L∞(M ), n ∈ N. Observe that (M ,E,m) obeys BE2(κn, N)for every n ∈ N. A priori, the Schrödinger operator ∆2κn is the form sum [42, p. 19]of ∆ and −2kn, the latter being viewed as self-adjoint [42, Thm. 1.7] multiplicationoperator on L2(M ) with domain D(−2κn) := f ∈ L2(M ) : kn f ∈ L2(M ) =L2(M ), n ∈ N. In fact [42, Prop. 3.1], ∆2κn is an operator sum, i.e. f ∈ D(∆) ifand only if f ∈ D(∆2κn) for every n ∈ N, and for such f ,

∆2κnf = ∆f − 2kn f m-a.e. (8.17)

Proposition 8.37. For every ω ∈ L2(T ∗M ) and every t ≥ 0,

|htω|2 ≤ p2κt

(|ω|2

)m-a.e.

Proof. We only concentrate on the nontrivial part t > 0. Let φ ∈ TestL∞(M ) benonnegative, and define F : [0, t] → R by

F (s) :=ˆ

M

φ p2κn

t−s

(|hsω|2

)dm =

ˆ

M

p2κn

t−sφ |hsω|2 dm,

where n ∈ N. As in the proof of Proposition 6.24, we argue that F is locallyabsolutely continuous on (0, t), and that for L

1-a.e. s ∈ (0, t),

F ′(s) = −ˆ

M

∆2κnp2κn

t−sφ |hsω|2 dm − 2ˆ

M

p2κn

t−sφ⟨hsω, ~∆hsω

⟩dm.

Given such an s ∈ (0, t), using a mollified version of (pt)t≥0 [87, p. 1648] we con-struct a sequence (fi)i∈N of nonnegative functions in TestL∞(M ) such that (fi)i∈N

and (∆fi)i∈N are bounded in L∞(M ), and fi → p2κn

t−sφ as well as ∆fi → ∆p2κn

t−sφpointwise m-a.e. as i → ∞. By Lebesgue’s theorem, (8.17), Remark 8.5 and (7.3),

F ′(s) = − limi→∞

ˆ

M

∆2κnfi |hsω|2 dm

− limi→∞

2ˆ

M

fi⟨hsω, ~∆hsω

⟩dm ≤ 0.

Integrating this inequality from 0 to t, employing the arbitrariness of φ and lettingn → ∞ via Levi’s theorem readily provides the claimed inequality.

As for Corollary 6.25, we have the following consequence of Proposition 8.37. Asimilar argument as for Lemma 8.39, providing an extension of Lemma 8.4 beyondregular vector fields which is needed for the proof of Theorem 8.40, is due to [16].

Corollary 8.38. For every ω ∈ D(~∆), there exists a sequence (ωn)n∈N in D(~∆) ∩L∞(T ∗M ) which converges to ω in H1,2(T ∗M ) such that in addition, ~∆ωn → ~∆ωin L2(T ∗M ) as n → ∞. If ω ∈ L∞(T ∗M ), this sequence can be constructed to beuniformly bounded in L∞(T ∗M ).

110 MATHIAS BRAUN

Lemma 8.39. For every n ∈ N, the conclusion from Lemma 8.4 holds for κ re-placed by κn as well as q = 1, ε > 0, X ∈ D(~∆)♯ ∩ L∞(TM ) and φ ∈ D(∆) ∩L∞(M ).

Proof. We shortly outline the argument. Let ψ ∈ Fb be nonnegative, and let(Xi)i∈N and (ψj)j∈N be sequences in Reg(TM ) and Test(M ) converging to X andψ in H1,2

♯ (TM ) and F, respectively. By Lemma 4.18, we may and will assume thatψj is nonnegative for every j ∈ N. Integrating Lemma 8.2, for κ replaced by κn,n ∈ N, against ψj and using that |Xi|2 ∈ Fb by Lemma 6.13, for every i, j ∈ N,

− 12

ˆ

M

⟨∇ψj ,∇|Xi|2

⟩dm −

ˆ

M

kn ψj|Xi|2

2dm

≥ˆ

M

ψj∣∣∇Xi

∣∣2

HSdm −

ˆ

M

ψj⟨Xi, (~∆X

i )♯⟩

dm.

Integrating by parts the last term, using Proposition 6.11 for the first, Lemma 8.8for the third and Theorem 7.5 and Lemma 7.15 for the last term, and finally inte-grating by parts back the last term we send i → ∞. This yields the previous inequal-ity for Xi replaced by X . Employing Lemma 8.8 and Proposition 6.11 again for thefirst term together with X ∈ L∞(TM ) and ∇ψj → ∇ψ in L2(TM ) as j → ∞, theabove estimate still holds for ψj replaced by ψ. Lastly, we insert ψ := φ [ϕ′

ε |X |2],ε > 0, where ϕε is defined as in Lemma 8.4 for q = 1. The term containing ϕ′′

ε |X |2coming from the Leibniz rule in the first integral cancels out with the third integralthanks to Lemma 6.13, and elementary further computations entail the claim.

Theorem 8.40 is known as Hess–Schrader–Uhlenbrock inequality [57, 58, 91] inthe case when M is a compact Riemannian manifold without boundary. A similar,analytic access to the latter on such M with convex boundary is due to [83, 89, 90].On general compact M with boundary, it has been derived in [67] using proba-bilistic methods. In the noncompact case without boundary, one can appeal toboth analytic [50] or stochastic [35, 75] methods. Moreover, recently, Lp-propertiesof (ht)t≥0 and related heat kernel estimates on Riemannian manifolds have beenstudied in [79] under Kato curvature conditions.

Theorem 8.40 (Hess–Schrader–Uhlenbrock inequality). For every ω ∈ L2(T ∗M )and every t ≥ 0,

|htω| ≤ pκt |ω| m-a.e.

Proof. Let (ωl)l∈N be a sequence in L1(T ∗M ) ∩ L∞(T ∗M ) which is obtained byappropriately cutting off and truncating the given ω. (In this case, truncationmeans multiplication with an indicator function of |ω| ≤ R, R > 0.) Moreover,given any ε > 0, define ϕε ∈ C∞([0,∞)) by ϕε(r) := (r + ε)1/2 − ε1/2. For anonnegative φ ∈ TestL∞(M ), given any l ∈ N, consider the function Fε : [0, t] → R

with

Fε(s) :=ˆ

M

φ pκn

t−s

(ϕε |hsωl|2

)dm =

ˆ

M

pκn

t−sφ[ϕε |hsωl|2

]dm.

As for Proposition 8.37, the function Fε is readily verified to be continuous on [0, t],locally absolutely continuous on (0, t), and integration and differentiation can beswapped in computing its derivative F ′

ε(s) at L1-a.e. s ∈ (0, t).

Given such an s ∈ (0, t), consider a sequence (fi)i∈N of nonnegative functions inTestL∞(M ) associated to p

κn

t−sφ as in the proof of Proposition 8.37. Then, accordingto Lemma 8.39 — since hsωl ∈ L∞(T ∗M ) thanks to Proposition 8.37 — (3.2) andLebesgue’s theorem,

F ′ε(s) = −

ˆ

M

∆κnpκn

t−sφ[ϕε |hsωl|2

]dm


− 2ˆ

M

pκn

t−sφ[ϕ′ε |hsωl|2

] ⟨hsωl, ~∆hsωl

⟩dm

= − limi→∞

ˆ

M

∆κnfi[ϕε |hsωl|2

]dm

− limi→∞

2ˆ

M

fi[ϕ′ε |hsωl|2

] ⟨hsωl, ~∆hsωl

⟩dm

≤ − limi→∞

2⟨κn

∣∣ fi |hsωl|2 ϕε |hsωl|2⟩

+ limi→∞

⟨κn

∣∣ fi ϕε |hsωl|2⟩ˆ

= −2⟨κn

∣∣ pκn

t−sφ |hsωl|2 ϕ′ε |hsωl|2

⟩+

⟨κn

∣∣ pκn

t−sφϕε |hsωl|2⟩.

ˆ

Integrating this inequality from 0 to t, sending ε → 0 with the aid of Lebesgue’stheorem and employing the arbitrariness of φ imply that, for every l, n ∈ N,

|htωl| ≤ pκn

t |ωl| m-a.e.

Sending l → ∞ and n → ∞ using Levi’s theorem terminates the proof.

Remark 8.41. Technical issues in general prevent us from proving Proposition 8.37or Theorem 8.40 beyond Assumption 8.36. The key reason is that integrated ver-sions or inequalities derived from (8.6) and (8.7) are hard to obtain beyond X ∈Reg(TM ) or integrands both belonging to Test(M ) and D(∆2κ).

It is outlined in Remark 8.5 above how to obtain more general versions underAssumption 8.36. The key obstacle, however, lies in dealing with the behavior of theterm in (8.6) containing the Hodge Laplacian or, in other words, to obtain an ana-logue to Lemma 8.39. In general, L∞-bounds for derivatives of p2κ

t−sφ or pκt−sφ, s ∈(0, t), lack for sufficiently many nonnegative φ ∈ L2(M ), but being able to integrateby parts this term essentially requires e.g. p2κ

t−sφ ∈ L∞(M ) and dp2κt−sφ ∈ L∞(T ∗M ).

(A related question is whether and when not only p2κt−sφ,∆

2κp2κt−sφ ∈ L∞(M ) —

which can always be achieved by [41, Sec. 6.1] — but also dp2κt−sφ ∈ L∞(T ∗M )

holds.) Compare with Proposition 7.8, Remark 7.9, (7.3) and Lemma 3.13. We alsocannot leave the Hodge Laplacian term as it is because we do not know if Reg(T ∗M )is dense in D(~∆) w.r.t. the induced graph norm. Under Assumption 8.36, these de-ductions could still be done thanks to the explicit relation (8.17) between D(∆2κ)and D(∆).

Remark 8.42. If we know that, given ω ∈ L2(T ∗M ), there exists a sequence (ωn)n∈N

in L2(T ∗M ) that L2-converges to ω such that htωn ∈ L∞(T ∗M ) for every t >0 and every n ∈ N, then Theorem 8.40 then Theorem 8.40 can be deduced bythe same arguments as above for more general κ ∈ K1−(M ). In particular, sincehtdfn ∈ L∞(T ∗M ) for every t > 0 and every n ∈ N, fn := maxminf, n,−n, byLemma 8.31, Theorem 8.40 recovers the gradient estimate [41, Thm. 6.9] for anyf ∈ F by (7.3).

On Riemannian manifolds with not necessarily convex boundary, such a sequencecan be constructed under further geometric assumptions [7, Thm. 5.2]. (The latterresult, in fact, implies Theorem 8.40 by Gronwall’s inequality.)

Appendix A. Extrinsic approaches

Lastly, we compare some recent extrinsic approaches [20, 23, 98] to boundaryobjects on RCD spaces with our intrinsic notions. More precisely, we outline somelinks to our notions of divergences and normal components.

A.1. Sets of finite perimeter. Let (M , d,m) be a locally compact RCD(K,∞)space, K ∈ R, with induced Dirichlet space (M ,E,m), cf. Example 1.19.

112 MATHIAS BRAUN

A.1.1. Identification of the measure-valued divergence. Following [23, Def. 4.1], letDMp(M ), p ∈ [1,∞], be the space of all X ∈ Lp(TM ) such that there existsdivX ∈ M

±f (M )E such that for every h ∈ Lipbs(M ),

−ˆ

M

h ddivX =ˆ

M

dh(X) dm.

The density of Lipbs(M ) in W 1,2(M ) [4] and [23, Prop. 4.6] yield the following.

Lemma A.1. Every X ∈ DM2(M ) belongs to D(div), and

divX = divX.

A.1.2. Gauß–Green formula. For boundary objects to really appear, we use theGauß–Green formulas for appropriate subsets E ⊂ M obtained in [23].

We say that a Borel set E ⊂ M has finite perimeter [23, Def. 3.3] if 1E ∈ BV(M ).It is associated with a Radon measure |D1E | ∈ M

+fR(M ) [23, Thm. 3.4] which is

supported on ∂E and, if m[∂E] = 0, in particular singular to m [23, Rem. 3.5].Here, the class of functions of bounded variation BV(M ) ⊂ L1(M ) can be definedin various ways [2, 23, 81] which all lead to the same spaces and objects in a largegenerality [2, Thm. 1.1]. (In particular, we require sets of finite perimeter to havefinite m-measure, although this is not strictly needed [20, Def. 1.1, Def. 1.2].)

We now make the following assumptions on E.

a. E satisfies the obstructions from Remark 1.20.b. The inclusion W 1,2(M )

∣∣E

⊂ W 1,2(E) from (1.6) is dense.c. E or Ec is a set of finite perimeter.

Item a. guarantees that (E, dE ,mE) induces a quasi-regular, strongly local Dirichletspace (E,EE ,mE), hence a tangent module L2(TE) w.r.t. mE . By (1.6), we canidentify L2(TM )

∣∣E

with L2(TE). Given anyX ∈ L2(TM ), denote byXE ∈ L2(TE)the image of 1EX under this identification. Of course, b. is satisfied if E has theextension property W 1,2(M )

∣∣E

= W 1,2(E), and if dE ≤ C d on E2 for some finiteC > 1. For a different variant of this condition b., see Section A.2 below.

Proposition A.2. For every X ∈ Test(TM ), there exists a unique 〈X, vE〉∂E ∈L∞(∂E, |D1E |) such that for every h ∈ W 1,2(E),

−ˆ

E

dh(X) dm =ˆ

E

hddivX

dmdm +

ˆ

∂E

h⟨X, vE

⟩∂E

d|D1E |. (A.1)

In particular, we have XE ∈ DTV(divE) with

divE XE =ddivX

dmmE-a.e.,

nE XE = −⟨X, vE

⟩∂E

|D1E |.

Proof. The last statement follows from (A.1), whence we concentrate on (A.1). ByLemma 3.15 and Lemma A.1, we have |divX | ≪ m. (And furthermore, divX hasfinite total variation.) Hence, under c., [23, Prop. 6.11, Thm. 6.13] implies thatthere exists a unique function 〈X, vE〉∂E ∈ L∞(∂E, |D1E |) such that (A.1) holdsfor every h ∈ Lipbs(M ). By b., E-quasi-uniform approximation [29, Thm. 1.3.3]and [23, Prop. 4.6], the latter extends to arbitrary h ∈ W 1,2(E).

Remark A.3. It would naturally follow from Proposition A.2 that the second fun-damental form of E at XE , X ∈ Test(TM ), according to Definition 8.23 — once(E,EE ,mE) is tamed by an appropriate κ ∈ K1−(M ) — is

IIE(XE , XE) =12

⟨∇|X |2, vE

⟩∂E

|D1E |


provided ∇|X |2 ∈ L∞(TM ). This latter assumption, however, seems to be quiterestrictive. (Compare with [47, Rem. 4.10, Rem. 4.11, Rem. 4.13].) Unfortunately,boundedness of the vector field in (A.1) seems to be essential in [23].

Remark A.4. The notation 〈X, vE〉∂E , X ∈ Test(TM ), in Proposition A.2 is purelyformal, in the sense that the authors of [23] neither consider any “tangent module”with scalar product 〈·, ·〉∂E over ∂E, nor define a unit normal vector field vE .

Example A.5. Another version of the Gauß–Green formula on RCD(K,N) spaces(M , d,m), K ∈ R and N ∈ [1,∞), has been obtained in [20, Thm. 2.2]. Retain theassumptions a., b. and c. on E ⊂ M . Then there exists a unique vE ∈ L2

E(TM ),the tangent module over ∂E [20, Thm. 2.1], with |vE | = 1 |D1E |-a.e. on ∂E suchthat for every X ∈ H1,2(TM ) ∩ D(div) ∩ L∞(TM ),

ˆ

E

divX dm = −ˆ

∂E

⟨trE(X), vE

⟩d|D1E |.

Here trE : H1,2(TM ) ∩ L∞(TM ) → L2E(TM ) is the trace operator over ∂E.

Replacing X by hX , where h ∈ W 1,2(M ) ∩ L∞(M ) has bounded support —recall Lemma 3.13 and Remark 6.10 — and using (3.3) as well as the arbitrarinessof h we obtain that XE ∈ D(divE) with

divE XE = divX mE-a.e.,

nE XE = −⟨X, vE

⟩|d1E |.

A.2. Regular semiconvex subsets. Consider the canonical Dirichlet space in-duced by an RCD(k, N) metric measure space (M , d,m), see Example 1.19, wherek : M → R is continuous and lower bounded as well as N ∈ [2,∞) [98, Def. 3.1,Def. 3.3, Thm. 3.4]. Let E ⊂ M be as in Remark 1.20 with m[E] < ∞.

For a function f on M or E, denote by fn its truncation maxminf, n,−n atthe levels n and −n, n ∈ N. Following [98, Def. 2.1] we set

W 1,1+(M ) :=f ∈ L1(M ) : fn ∈ F for every n ∈ N,

supn∈N

∥∥|fn| + |dfn|∥∥

L1(M)< ∞

.

Let W 1,1+(E) be defined analogously w.r.t. W 1,2(E) and |d · |E . We assume thatE has regular boundary [98, p. 1702], i.e. v ∈ D(∆) with v,∆v ∈ C(M ) ∩ L∞(M ),where v := d(·, E) − d(·, Ec) is the signed distance function from ∂E, and

W 1,1+(M )∣∣E

= W 1,1+(E).

Then thanks to [98, Lem. 6.10], there exists a nonnegative σ ∈ M+f (M )E supported

on ∂E such that for every h ∈ W 1,2b (M ),

ˆ

∂E

hdσ =ˆ

E

dv(∇h) dm +ˆ

M

∆v h dm. (A.2)

Lemma A.6. In the notation of Subsection A.1.2, the vector field (∇v)E ∈ L2(TE)belongs to DL2 (divE) with

divE(∇v)E = ∆v mE-a.e.,

nE(∇v)E = σ.

Proof. Since m[E] < ∞, any given h ∈ W 1,2bc (E) belongs to W 1,1+(E), and hence

to W 1,1+(M )∣∣E

by regularity of ∂E. Thus, there exists h ∈ W 1,1+(M ) such thath = h m-a.e. on E. In particular, hn ∈ W 1,2

b (M ) for every n ∈ N. Since hn = hm-a.e. on E for large enough n ∈ N, the claim follows from (A.2).

114 MATHIAS BRAUN

Remark A.7. If the integration by parts formula as in [98, Lem. 6.11] holds —which, in fact, uniquely characterizes σ — a similar argument as for Lemma A.6yields that for every f ∈ D(∆) with ∇f ∈ L∞(TM ), we have (∇f)E ∈ D(divE)with

divE(∇f)E = ∆f mE-a.e.,

nE(∇f)E = 〈∇f,∇v〉∼ σ.

The latter is well-defined thanks to Lemma 6.13 and Corollary 5.12.

Proposition A.8. Given any X ∈ Reg(M ), define X⊥ ∈ L2(TM ) by

X⊥ := 〈X,∇v〉 ∇v.Then X⊥

E ∈ D(divE) with

divE X⊥E = 〈X,∇v〉 ∆v + ∇X : (∇v ⊗ ∇v) mE-a.e.,

nE X⊥E = 〈X,∇v〉∼ σ.

Proof. Observe that 〈X,∇v〉 ∈ W 1,2b (M ), so that both the statements make sense.

The claimed formulas follow from Lemma A.6 as well as Lemma 3.13 while notingthat by Proposition 6.11 and since |∇v| = 1 m-a.e.,

⟨∇〈X,∇v〉,∇v

⟩= ∇X : (∇v ⊗ ∇v) + Hess v(X,∇v)

= ∇X : (∇v ⊗ ∇v) + d|∇v|2(X)/2

= ∇X : (∇v ⊗ ∇v) m-a.e.

Remark A.9. We do not know if the pointwise defined second fundamental formfrom [98, Rem. 5.13] is related to IIE . Similarly to Remark A.3, by Proposition A.8these notions coincide if X ∈ Reg(TM ) obeys |X |2 ∈ D(Hess), but we do not knowif many of such X can be found in general.

At least, this time the taming condition for (E,EE ,mE) is already provided onceone can verify that the taming distribution

κ := −k−mE − l

− σ,

for some appropriate l ∈ C(M ), according to [98, Thm. 6.14] and [41, Prop. 2.16]belongs to K1−(E). See [18, 41] for examples in this direction.

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University of Bonn, Institute for Applied Mathematics, Endenicher Allee 60, 53115Bonn, Germany

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