LOCAL T THEOREMS AND APPLICATIONS IN...

LOCAL T (b) THEOREMS AND APPLICATIONS IN PDE

STEVE HOFMANN

Abstract. A Tb theorem is a boundedness criterion for singular inte-grals, which allows the L2 boundedness of a singular integral operatorT to be deduced from sufficiently good behavior of T on some suitablenon-degenerate test function b. However, in some PDE applications, in-cluding, for example, the solution of the Kato problem for square rootsof divergence form elliptic operators, it may be easier to test the opera-tor T locally (say on a given dyadic cube Q), on a test function bQ thatdepends upon Q, rather than on a single, globally defined b. Or to bemore precise, in the applications, it may be easier to find a family ofbQ’s for which TbQ is locally well behaved, than it is to find a single b forwhich Tb is nice globally. In these lectures, we’ll discuss some versionsof local Tb theorems, as well as some applications to PDE.

1. Lecture 1. Introduction: Boundedness criterion for SIO’sand square functions

The Tb Theorem, and its predecessor, the T1 Theorem, were introducedin large part to better understand the Cauchy integral operator on a Lip-schitz curve, and the related Calderon commutators. In these lectures, weshall discuss more recent “local” versions of the Tb Theorem, as well as theapplication of such theorems to some questions in PDE.

We begin by recalling the statements of the original T1 and Tb theorems.To this end, we require a few definitions.

1.1. Calderon-Zygmund Kernel definition. A “standard” Calderon-Zygmund (CZ ) kernel is a function K(x, y) defined on Rn × Rn \ x = ysuch that

(i) |K(x, y)| ≤ C

|x− y|n

and

(ii) |K(x, y)−K(x+ h, y)|+ |K(x, y)−K(x, y + h)| ≤ C|h|α

|x− y|n+α,

for some α > 0, if |h| ≤ |x−y|2 .

2000 Mathematics Subject Classification. Primary 42B20.Key words and phrases. Tb theorems, singular integrals, square functions, elliptic PDE.Research supported in part by a grant from the NSF.

1

2 STEVE HOFMANN

Following Coifman and Meyer [CM], we have:

1.2. Singular Integral Operator definition. A singular integral operator(SIO) is a mapping T : C∞

0 → (C∞0 )′ associated to a standard CZ kernel in

the sense that

〈Tf, g〉 =∫ ∫

K(x, y)f(y) dy g(x) dx

for any f, g ∈ C∞0 with disjoint supports.

For such operators, it is a classical result that L2 bounds imply Lp bounds,1 < p <∞. The fundamental result is the following:

1.3. Theorem. [Calderon-Zygmund, 1952] If an SIO T : L2 → L2, thenT : Lp → Lp, for 1 < p <∞, and it is of weak-type (1, 1).

Moreover, Peetre, Spanne, and Stein proved independently that:

1.4. Theorem. [Peetre [P]/Spanne [Sp]/Stein [St], 1966] If an SIOT : L2 → L2, then T : L∞ → BMO.

We recall that BMO is the space of locally integrable functions moduloconstants for whom the norm

‖b‖∗ = sup |Q|−1

∫Q|b(x)− [b]Q|dx

is finite. Here, the supremum runs over all cubes (balls work just as well)with sides parallel to the co-ordinate axes, and

[b]Q := |Q|−1

∫Qb(x)dx =:

∫Qb(x)dx.

Thus, the fundamental issue for SIOs is whether T : L2 → L2.In the convolution case, by Plancherel’s theorem, we need precisely that

K ∈ L∞, where

K(ξ) = limε→0

∫ε<|x|< 1

ε

e2πiξ·xK(x) dx .

In the general (not necessarily convolution) case, the prototypical result isthe criterion of David and Journe [DJ].

1.5. T1 Theorem. [David-Journe, 1984] Let T be an SIO. Then T :L2 → L2 if and only if T1 ∈ BMO, T ∗1 ∈ BMO, and T satisfies the weakboundedness property (WBP).

The latter notion is defined as follows:

(1.6) sup(R−n|〈Tϕ, ψ〉|

)≤ C

where the supremum runs over all R > 0, all balls B = B(x,R), and allϕ,ψ ∈ C∞

0 (B) such that ‖ ϕ ‖∞ +R ‖ ∇ϕ ‖∞≤ 1, ‖ ψ ‖∞ +R ‖ ∇ψ ‖∞≤ 1.Let us make a few comments on the previous theorems.

LOCAL T (b) THEOREMS 3

1.7. Remark. In practice, the WBP (1.6) is usually easier to verify than thehypotheses T1 ∈ BMO, T ∗1 ∈ BMO. E.g. if T is a principal value operatorassociated to an antisymmetric kernel K(x, y) (i.e. K(x, y) = −K(y, x)),then WBP holds automatically.

Also, it is not hard to show that T1 ∈ BMO, T ∗1 ∈ BMO and WBP allfollow from the “local” T1 condition

supQ

∫Q

(|T 1Q |+ |T ∗ 1Q |) ≤ C ,

or its “smoothly truncated” analogue:

supQ

∫Q

(|T ηQ |+ |T ∗ ηQ |

)≤ C ,

where ηQ ∈ C∞0 (3Q), with ηQ ≡ 1 on 2Q. These two estimates are equiv-

alent, since the size condition 1.1 (i) and a version of Hardy’s inequalityimply that the error between the two averages on Q is always bounded.

1.8. Remark. The fact that L2 boundedness implies the WBP (1.6) is aneasy consequence of Cauchy-Schwarz. Also, the fact that L2 boundednessimplies that T1 ∈ BMO, T ∗1 ∈ BMO is just a consequence of Theorem1.4.

Hence, the main direction in the Theorem 1.5 is the direction “⇐=”.

1.9. Remark. Note the following converse to Theorem 1.4. Suppose T :L∞ → BMO and T ∗ : L∞ → BMO. By the work of Fefferman and Stein[FS], it follows by duality that T : H1 → L1, and also then by interpolationthat T : L2 → L2.

Hence, the T1 theorem says that T and T ∗ need not be tested on all ofL∞ in order to be able to conclude that T : L2 → L2, but only on theconstant function 1 (assuming WBP (1.6)), or else, locally on 1Q.

The T1 Theorem 1.5 is powerful enough to yield almost immediatelyCalderon’s Theorem [C1] on the L2 boundedness of the first “Calderon com-mutator”:

C1 f(x) = p.v. c

∫R

A(x)−A(y)(x− y)2

f(y) dy = ([DH,A]f) (x) ,

where x ∈ R, D = ddx , H is the Hilbert transform, and (abusing nota-

tion slightly) A denotes both a Lipschitz function, and also the operator ofmultiplication by the function A.

Indeed, since A is Lipschitz, the kernel K(x, y) = A(x)−A(y)(x−y)2

is standard.Moreover, K(x, y) = −K(y, x) (so we have the WBP (1.6), and also thatC∗

1 = −C1), hence we need to check only that C1 1 ∈ BMO.Now formally, since DH = HD, we have

[DH,A] 1 = DHA−ADH 1 = HDA = HA′ ∈ BMO,

by Theorem 1.4, since A′ ∈ L∞.

4 STEVE HOFMANN

By induction, this reasoning applies to the higher order commutators

Ck f(x) = p.v. c

∫R

(A(x)−A(y))k

(x− y)k+1f(y) dy ,

and hence, for ‖ A′ ‖∞ small, to the Cauchy integral

CAf(x) = p.v. c

∫R

1(x− y) + i (A(x)−A(y))

f(y) dy ,

by expanding the kernel in a power series in A(x)−A(y)x−y (hence we obtain an

easy alternative proof of Calderon’s theorem from 1977 [C2].)For arbitary (i.e., not just small) Lipschitz constant, the L2 boundedness

of CA was obtained by Coifman, McIntosh and Meyer [CMcM].Observe now that if γ is the Lipshitz curve x→ x+ iA(x), then

CA(1 + iA′) = c

∫γ

1z − v

dv =12,

by Plemelj’s formula (only formally, since γ is an unbounded graph, butsome version of this can be made rigorous.) Consequently, the theorem of[CMcM] may be recovered from the following

1.10. Tb Theorem. [David-Journe-Semmes [DJS], 1985] Suppose thereexist “accretive” functions b1, b2, i.e.

bi ∈ L∞, Re(bi) ≥ c > 0, i = 1, 2,

such that b2Tb1 satisfies the WBP (1.6), and both Tb1 and T ∗b2 are in BMO(the special case Tb1 = 0 = T ∗b2 was treated a little earlier by McIntosh andMeyer [McM]). Then T : L2 → L2.

1.11. Remark. Observe that 1 + iA′ is accretive.

1.12. Remark. In some applications, finding a globally defined function bon which T behaves well may be harder than finding a family of functionsbQ, indexed on the dyadic cubes Q, such that TbQ behaves well on Q.This leads to the notion of a “Local Tb Theorem”, the first version of whichwas obtained by M. Christ [Ch].

1.13. Local Tb Theorem. [Christ, 1990] Let T be an SIO, with K(x, y) ∈L∞ (qualitatively.) Suppose that there exist C0 <∞, δ > 0, and two systemsof functions b1Q, b2Q, indexed by the dyadic cubes, such that biQ issupported in Q, for i = 1, 2, and

(i) ‖ biQ ‖∞≤ C0, i = 1, 2

(ii) ‖ Tb1Q ‖L∞(Q) + ‖ T ∗b2Q ‖L∞(Q)≤ C0

(iii) δ ≤∣∣∣∣∫

QbiQ

∣∣∣∣ , i = 1, 2.


Then T : L2 → L2.

1.14. Remark. The L∞ hypothesis on K(x, y) is merely qualitative: the L∞

bounds do not appear in the estimates for ‖ T ‖2→2. For example, thetheorem applies to (smoothly) truncated singular integrals, for which thegoal is to obtain L2 bounds independent of the truncation.

1.15. Remark. The above Theorem 1.13 is valid in the setting of “spaces ofhomogeneous type”, since in that context Christ has established the exis-tence of a “dyadic cube” structure [Ch].

1.16. Remark. Extensions of either the global or local Tb theorems, to thesetting of “non-doubling” measures, have been given by G. David [D] andNazarov-Treil-Volberg [NTV1, NTV2].

1.17. Remark. Christ’s theorem 1.13 is related to the theory of analyticcapacity and to the Painleve problem, the solution of which (by Melnikov,Mattila and Verdera [MMV] and G. David [D] in special cases, and by Tolsain the general case [T]; see also the work of Volberg [Vo] for the higherdimensional analogue) involved the use of Tb theorems.

In these lectures we consider extensions of Christ’s result 1.13 in whichwe allow weaker control (say scale invariant Lp bounds instead of L∞) forbQ and TbQ, and we will discuss applications to elliptic PDEs (including theKato problem and bounds for layer potentials.)

1.1. T1 and Tb theorems for square functions (Christ/Journe [CJ] &Semmes [S]). Following [CJ], we say that a family of kernels ψt(x, y)t∈R+

is a “standard Littlewood-Paley” (LP) family if the following two conditionshold:

(1.18) |ψt(x, y)| ≤ Ctα

(t+ |x− y|)n+α , for some α > 0,

(1.19) |ψt(x, y + h)− ψt(x, y)| ≤ C|h|α

(t+ |x− y|)n+α , for |h| < t.

These bounds are satisfied, e.g., by t times the gradient of the Poisson kernelfor the half space.

1.20. T1 Theorem for Square Functions. [Christ-Journe [CJ]] Let

θtf(x) :=∫ψt(x, y)f(y) dy ,

and assume that ψt is an LP family, (i.e. (1.18), (1.19) hold). Suppose that

|θt 1(x)|2 dxdtt

is a Carleson measure. Then the following estimate holds

(1.21)∫ ∞

0

∫Rn

|θtf(x)|2 dxdtt

≤ C‖f‖22

6 STEVE HOFMANN

Recall that a measure µ in Rn+1+ = Rn×(0,∞) is a Carleson measure if for

any cube Q ⊂ Rn, we have that µ(RQ) ≤ C|Q|, where RQ = Q× (0, `(Q)) ⊂Rn+1

+ . Here `(Q) denotes the sidelength of the cube Q. So, µ acts, in somesense, like an n dimensional measure, even though it is an (n+1) dimensionalmeasure.

1.22. Remark. The converse is due essentially to Fefferman and Stein [FS].

Proof. (Sketch of proof of Theorem 1.20) Following Coifman and Meyer[CM], we write

θt = [θt − (θt 1)Pt] + (θt 1)Pt = Rt + (θt 1)Pt ,

where Pt is a “nice” approximation to the identity. The term (θt 1)Pt isok by hypothesis and Carleson’s lemma, whereas Rt is ok by orthogonalityarguments, since Rt 1 = 0, and the kernel of Rt is also of standard LP type.

1.23. Tb Theorem for Square Functions. [Semmes [S]] Let

θtf(x) :=∫ψt(x, y)f(y) dy ,

and assume that ψt satisfies LP (i.e. (1.18), (1.19)). Suppose there existsan accretive function b such that

|θtb(x)|2dxdt

t

is a Carleson measure. Then the square function estimate (1.21) holds.

Proof. (Sketch of proof of Theorem 1.23) By Theorem 1.20, it is enough toshow that |θt 1(x)|2 dxdt

t is a Carleson measure. By accretivity, we have that

|θt 1 | ≤ C| (θt 1)Ptb| .Localizing, we may assume without loss of generality, that b ∈ L2. Againfollowing [CM], we can write

(θt 1)Ptb = −Rtb+ θtb ,

where Rt is as above, and is therefore still under control. The term θtb is okby hypothesis.

1.24. Remark. Observe that this argument carries over if b varies with Q,i.e. if we have a system bQ satisfying

(1.25) supQ

∫Q

∫ `(Q)

0|θt 1 |2 dxdt

t≤ C sup

Q

∫Q

∫ `(Q)

0| (θt 1)PtbQ|2

dxdt

t

and ∫ `(Q)

0

∫Q|θtbQ|2

dxdt

t≤ C .


This observation is essentially due to Auscher and Tchamitchian [AT], andis the starting point for the solution of the Kato problem.

Thus, it is natural to pose the question: when does (1.25) hold? Infact, the solution to the Kato problem provided a sufficient condition whichanswers this question (see in particular (1.27), (1.29) and (1.36), (1.38)below), and this is the real essence of the next two theorems.

1.26. Local Tb Theorem for Square Functions. [essentially [HMc],[HLMc], [AHLMcT]; see also [A] and [H]] Suppose ψt(x, y) is a standardLP family (i.e. satisfying (1.18), (1.19)), and θtf(x) :=

∫Rn ψt(x, y)f(y) dy .

Suppose also that there exist δ > 0, C0 < ∞ such that for any dyadic cubeQ, there exists a function bQ satisfying

(1.27)∫

Rn

|bQ|2 ≤ C0|Q|

(1.28)∫ `(Q)

0

∫Q|θtbQ(x)|2 dxdt

t≤ C0|Q| .

(1.29) δ ≤∣∣∣∫

QbQ

∣∣∣ .Then the estimate (1.21) holds:∫ ∞

0

∫Rn

|θtf(x)|2 dxdtt

≤ C ‖ f ‖22 .

Moreover, if in addition we have that

(1.30) |ψt(x+ h, y)− ψt(x, y)| ≤ C|h|α

(t+ |x− y|)n+α , for |h| < t ,

then we may relax both (1.27) and (1.28) to the following two conditions

(1.31)∫

Rn

|bQ|q ≤ C0|Q|

and

(1.32)∫

Q

(∫ `(Q)

0|θtbQ(x)|2 dt

t

) q2

dx ≤ C0|Q| ,

for some q > 1.

Before sketching the proof of Theorem 1.26, let us state a lemma. Werecall that for a cube Q, RQ := Q × (0, `(Q)) is the associated “Carlesonbox”.

8 STEVE HOFMANN

1.33. Lemma. [“John-Nirenberg” lemma for Carleson measures].Let µ be a measure in Rn+1

+ . Suppose that there exists η > 0 such that forevery dyadic cube Q, there is a collection Qj of non-overlapping dyadicsub-cubes of Q, satisfying

|EQ| := |Q \ (∪Qj) | ≥ η|Q| ,

and for which the “η-ample sawtooth” ΩQ := RQ \(∪RQj

)satisfies

µ (ΩQ) ≤ C1|Q| .Then µ is a Carleson measure, and

µ (RQ) ≤ C1

η|Q| .

One may prove Lemma 1.33 by iterating and summing a geometric series.

Proof. (Sketch of proof of Theorem 1.26 for q = 2) By Theorem 1.20 it isenough to show that |θt 1(x)|2 dxdt

t is a Carleson measure. By the John-Nirenberg Lemma 1.33, it is enough to control |θt 1(x)|2 dxdt

t in an amplesawtooth (with fixed η) for each dyadic cube Q.

In order to establish a version of (1.25) (see Theorem 1.23 and Remark1.24), our goal is to show that

(1.34) |θt 1(x)| ≤ C|θt 1AtbQ(x)| ,on an ample sawtooth. Here, At denotes the dyadic averaging operator, i.e.Atf(x) =

∫Q(x,t) f , where Q(x, t) is the minimal dyadic cube that contains

x satisfying `(Q(x, t)) ≥ t.Indeed, suppose that (1.34) holds. Then, as in [CM],

(θt 1)AtbQ = ((θt 1)At − θt) bQ + θtbQ = −RtbQ + θtbQ .

The operator Rt satisfies (1.21), so it is ok, and the term θtbQ is ok byhypothesis (1.28).

We will establish (1.34) via a stopping time argument. Indeed, by renor-malizing, let us assume that δ = 1, i.e. that

∫Q bQ = 1. Select now those

dyadic subcubes Qj of Q that are maximal with respect to the property that∣∣∣∫Qj

bQ

∣∣∣ < 12.

Let us now set ΩQ = RQ \(∪RQj

). By construction, if (x, t) ∈ ΩQ, then∣∣∣AtbQ

∣∣∣ ≥ 12 . Hence, (1.34) holds with C = 2 in ΩQ.

We are thus left to show that ΩQ is “ample”. Set E := Q \ (∪Qj). Then

|Q| =∫

QbQ =

∫EbQ +

∑∫Qj

bQ


≤ |E|12

(∫b2Q

) 12

+12

∑|Qj | ≤

√C0|E|

12 |Q|

12 +

12|Q| ,

therefore |Q| ≤ 4C0|E|.

We defer for now the proof of the Lq version of Theorem 1.26, and proceedto extend the L2 case to the vector-valued setting, which yields a toy versionof the proof of the Kato conjecture.

1.35. Theorem. Let ~θt and ~ψt be as above (e.g. as in Theorem 1.26), exceptthat now they are CN -valued. Suppose that there exist constants δ and C0

such that for any dyadic cube Q, there exists an N ×N (complex) matrix-valued mapping BQ(x) satisfying the following three conditions:

(1.36)∫

Rn

|BQ|2 ≤ C0|Q|

(1.37)∫ `(Q)

0

∫Q|~θtBQ(x)|2 dxdt

t≤ C0|Q| ,

where here(~θtBQ

)j

= θitB

ijQ ; and, for all unit vectors ν ∈ CN ,

(1.38) δ ≤ Re ν ·(∫

QBQ

)ν .

Then the estimate corresponding to (1.21) holds:∫ ∞

0

∫Rn

|~θt~f(x)|2 dxdt

t≤ C ‖ ~f ‖2

2 ,

where ~θt~f = θi

tfi.

Proof. (Sketch of proof of Theorem 1.35) By Theorem 1.20 it is enough toshow that |~θt1(x)|2 dxdt

t :=∑N

i=1 |θit 1(x)|2 dxdt

t is a Carleson measure.For a fixed ε > 0 to be chosen later, cover CN ≈ R2N by K = K(ε,N)

cones of aperture ε of the form

Γε := z ∈ CN :∣∣∣ z|z| − ν

∣∣∣ < ε ,

where |ν| = 1. We can then write

|~θt1(x)|2 =K∑

j=1

|~θt1(x)|2 1Γεj

(~θt1).

10 STEVE HOFMANN

Since K is bounded, it is enough to consider each cone separately and showthat, for each j,

supQ

∫ `(Q)

0

∫Q|~θt1(x)|2 1Γε

j

(~θt1) dxdt

t≤ C .

As before, we show that, in a fixed cone,

|~θt1(x)| ≤ C∣∣∣(~θt1

)AtBQ(x)

∣∣∣ .Renormalizing, we may assume that δ = 1. Once again, we shall performa stopping time argument to produce an ample sawtooth in which we haveappropriate accretivity. To wit: we extract a collection Qj of dyadicsubcubes of Q which are maximal with respect to the property that at leastone of the following two estimates holds:

(1.39)∫

Qj

|BQ| ≥14ε

,

or

(1.40) Re ν ·

(∫Qj

BQ

)ν ≤ 3

4.

As in the scalar version, we have that E := Q\(∪Qj) satisfies |E| > η|Q| forsome uniform η > 0. Indeed, if M denotes the Hardy-Littlewood maximaloperator, then∑

(1.39) holds

|Qj | ≤ |M(BQ) >14ε| ≤ Cε2

∫|BQ|2 ≤ Cε2|Q| .

Consequently,

|Q| ≤ Re ν ·(∫

QBQ

)ν

≤∫

E|BQ|+ Re

∑Qj :(1.40) holds

ν ·

(∫Qj

BQ

)ν +

∑Qj :(1.39) holds

∫Qj

|BQ|

=: I + II + III

Now notice thatII ≤ 3

4

∑|Qj | ≤

34|Q| ,

andIII ≤

∫M(BQ)> 1

4ε|BQ| ≤ Cε|Q|

12 ‖ BQ ‖2≤ Cε|Q|.

Choosing ε small enough, we have that

18|Q| ≤ |E|

12 ‖ BQ ‖2≤ C|E|

12 |Q|

12 ,


and thus, |E| > η|Q| as desired. Moreover, if z ∈ Γε, and if (x, t) ∈ ΩQ =RQ \ (∪Qj), then∣∣∣ z|z| ·AtBQ(x)ν

∣∣∣ ≥ ∣∣∣ν ·AtBQ(x)ν∣∣∣− ∣∣∣( z

|z|− ν

)·AtBQ(x)ν

∣∣∣≥ 3

4− ε

14ε

=12.

Taking z = ~θt1(x), we have for (x, t) ∈ ΩQ,∣∣∣~θt1(x)∣∣∣ ≤ 2

∣∣∣~θt1(x) ·AtBQ(x)ν∣∣∣ ≤ 2

∣∣∣~θt1(x)AtBQ(x)∣∣∣

Now continue as in the scalar case, i.e. as in the proof of Theorem 1.26.

2. Lecture 2: Application to the Kato problem and layerpotentials

Suppose A(x) is an n× n, L∞, complex matrix defined on Rn, satisfyingthe following accretivity (i.e., ellipticity) condition: there exists λ > 0 suchthat

λ|ξ| ≤ Re 〈A(x)ξ, ξ〉, for all ξ ∈ Cn .

In [AHLMcT] (but see also [HMc], [HLMc], [AHLT]), it is proved thatthere exists a C = C(n, λ, ‖ A ‖∞) such that

2.1. Kato Estimate. ‖√−∇ ·A∇u ‖2 ≤ C ‖ ∇u ‖2 .

By “T1” type reasoning (in the spirit of Christ-Journe, Theorem 1.20),the proof of the Kato estimate 2.1 may be reduced to showing that

(2.2) |~θt1(x)|2 dxdtt

is a Carleson measure,

where ~θt = te−t2LdivA, and L = −divA∇. Roughly speaking, one proves(2.2) by applying (a variant of) the vector-valued version of the local Tbtheorem for square functions, Theorem 1.35, with N = n and

BQ = ∇(e−ε2`(Q)2LϕQ

),

where ε > 0 is small, but fixed depending on n, λ, ‖ A ‖∞, and ϕQ(x) :=(x− xQ)ηQ, with ηQ ∈ C∞

0 (5Q), and ηQ ≡ 1 on 4Q.

2.3. Remark. Even though ~θt does not have a “standard LP” kernel, nonethe-less similar reasoning to the proof of Theorem 1.35 still applies. Hypotheses(1.36), (1.37), and (1.38) of the theorem are obtained respectively, for thepresent choice of BQ, from (2.4), (2.5) and (2.6) below:

(2.4) ellipticity, integration by parts, and the fact that ϕQ is Lipschitz,

12 STEVE HOFMANN

(2.5)∫

Rn

∣∣∣divABQ

∣∣∣2 ≤ C

ε2 (`(Q))2|Q|, 1

and for some β > 0,

(2.6)∫

QBQ = 1n×n +O(εβ) ,

where 1n×n denotes the n×n identity matrix. Indeed, that (2.4) =⇒ (1.36)(with constant independent of ε) is fairly routine (along the lines of the proofof Caccioppoli’s inequality), and that (2.6) =⇒ (1.38) is obvious. Moreover,semigroup bounds plus (2.5) imply∫ ∣∣∣te−t2LdivABQ

∣∣∣2 dx ≤ C

ε2

(t

`(Q)

)2

|Q| ,

from which (1.37) follows easily.In turn, (2.5) holds as a consequence of the analyticity of the semigroup,

and (2.6) holds, at least in the case that the semigroup satisfies a pointwiseGaussian upper bound, because for Lipschitz ϕ we then have

‖e−ε2`(Q)2Lϕ− ϕ‖∞ ≤ C‖∇ϕ‖∞ ε `(Q),

whence by integrating we obtain∣∣∣∣∣∫

Q∇(e−ε2`(Q)2LϕQ − ϕQ

)∣∣∣∣∣ ≤ Cε .

Now, in Q, ∇ϕQ = 1n×n, and (2.6) follows, in this case with β = 1. Theproof when Gaussian bounds are lacking is slightly more involved: we referthe reader to [AHLMcT] for the details in that case.

We now sketch the proof of the Lq version of Theorem 1.26.

Proof. (Sketch of proof of the Lq version of Theorem 1.26) Again, by The-orem 1.20, it is enough to show that

(2.7) |θt 1(x)|2 dxdtt

is a Carleson measure.

2.8. Lemma. [Lq version of John-Nirenberg lemma] Suppose there areconstants η > 0, and C1 <∞ such that for any dyadic cube Q, there existsa collection Qj of non-overlapping dyadic subcubes of Q, satisfying

(2.9)∑

|Qj | ≤ (1− η)|Q| ,

and

(2.10)∫

Q

(∫ `(Q)

τQ(x)|θt 1(x)|2 dt

t

) q2

dx ≤ C1|Q| ,

1For ε = 1, this is essentially the condition that appears in [AT] and which leads to(1.37).


where τQ(x) :=∑

1Qj (x)`(Qj), (i.e. the graph of the step function τQ isthe “horizontal” part of the boundary of ΩQ = RQ \

(∪RQj

).) Then (2.7)

holds.

We defer momentarily the proof of Lemma 2.8.Recall that we are assuming (1.31), (1.32), and (1.29). We claim that it is

enough to show that for each Q, there exists a collection of non-overlappingdyadic subcubes Qj satisfying (2.9) such that

(2.11)∫Q

(∫ `(Q)


t

) q2

dx ≤ C

∫Q

(∫ `(Q)

τQ(x)| (θt 1)AtbQ|2

dt

t

) q2

dx .

Indeed, as usual, following [CM],

(θt 1)AtbQ = [(θt 1)At − θt] bQ + θtbQ =: RtbQ + θtbQ ;

the operator Rt gives a bounded square function on Lq, hence it is ok by(1.31), whereas the term θtbQ is ok by (1.32). Now apply the Lq John-Nirenberg Lemma 2.8. In turn, to get (2.11), we just use the same stoppingtime argument as in the case q = 2.

Thus, the key now is to prove the Lq John-Nirenberg Lemma 2.8. This,in turn, will follow from a “weak-type John-Nirenberg Lemma” that we nowstate:

2.12. Lemma. [Weak-type version of John-Nirenberg lemma; essen-tially appears in Auscher-Hofmann-Lewis-Tchamitchian [AHLT],earlier BMO version due to F. John] Suppose there exist N < ∞,β ∈ (0, 1) such that, for any dyadic cube Q,

(2.13)∣∣∣x ∈ Q : gQ(x) > N

∣∣∣ ≤ (1− β) |Q| ,

where

gQ(x) :=

(∫ `(Q)

0|θt 1(x)|2 dt

t

) 12

.

Then (2.7) holds, i.e.

(2.14) |θt 1(x)|2 dxdtt

is a Carleson measure.

We again defer the proof of Lemma 2.12 momentarily, and use it to provethe Lq John-Nirenberg Lemma 2.8.

Proof. (Sketch of proof of the Lq John-Nirenberg Lemma 2.8) Recall thatwe are assuming (2.9) and (2.10), and that τQ(x) = 0 for x ∈ E.

Set GN := x ∈ Q : gQ(x) > N and E := Q \ (∪Qj). Then,

14 STEVE HOFMANN

∣∣∣GN

∣∣∣ ≤∑ |Qj |+∣∣∣x ∈ E : gQ(x) > N

∣∣∣≤ (1− η)|Q|+

∣∣∣∣∣x ∈ Q :

(∫ `(Q)


t

) 12

> N

∣∣∣∣∣(2.15)

≤ (1− η)|Q|+ C1

N q|Q| ≤ (1− η

2)|Q|(2.16)

where in (2.15) we have used (2.9) and the fact that τQ(x) = 0 for x ∈ E,and in (2.16) we have used (2.10) and Tchebychev, and the last inequalityholds if N is chosen large enough.

Hence the hypotheses of the weak-type John-Nirenberg Lemma 2.12 hold,so (2.14) holds.

We now prove the weak-type John-Niremberg Lemma 2.12.

Proof. (Sketch of proof of the weak-type John-Niremberg Lemma 2.12). Set

K(ε) := supQ

1|Q|

∫Qg2Q,ε ,

where

gQ,ε :=

(∫ min(`(Q), 1ε)

ε|θt 1(x)|2 dt

t

) 12

(and := 0 if `(Q) ≤ ε.) We want to show that

sup0<ε<1

K(ε) <∞ .

To that end, setGN,ε := x ∈ Q : gQ,ε(x) > N ,

and recall that ψt(x, y) is Holder continuous in x, hence gQ,ε(x) is continuous,and GN,ε is open. Consequently, we have the Whitney decomposition

GN,ε = ∪Qk .

Set now FN,ε := Q \GN,ε. Then

(2.17)∫

Qg2Q,ε =

∫FN,ε

g2Q,ε +

∑k

∫Qk

g2Q,ε

≤ N2|Q|+∑

k

∫Qk

g2Qk,ε +

∑k

∫Qk

∫ min(`(Q), 1ε)

max(`(Qk),ε)|θt 1(x)|2 dx dt

t

=: N2|Q|+ I + II .


Now, by (2.13),

I ≤ K(ε)∑

|Qk| ≤ K(ε)(1− β)|Q| ,

so that this term may be hidden on the left-hand side of the estimate forK(ε) that we are setting up in (2.17). Thus, it is enough to show that

(2.18) II ≤ C∑

|Qk| ≤ C|Q| .

To prove (2.18), since Qk is a Whitney cube, there exists xk ∈ FN,ε suchthat dist(xk, Qk) ≤ C`(Qk). Then,

∫Qk

∫ min(`(Q), 1ε)

max(`(Qk),ε)|θt 1(x)|2 dx dt

t

.∫

Qk

∫ Cn`(Qk)

`(Qk)|θt 1(x)|2 dt

tdx

+∫

Qk

∫ `(Q)

Cn`(Qk)|θt 1(x)− θt 1(xk)|2

dt

tdx + |Qk|g2

Q,ε(xk)

=: A+ B + C .

Since xk ∈ FN,ε,C ≤ N2|Qk| .

Also, by Holder continuity of ψt(x, y) in the x variable, we have that

B . |Qk|∫ ∞

Cn`(Qk)

(`(Qk)t

)α dt

t= C|Qk| ,

if Cn is chosen large enough that |x− xk| < Cn`(Qk) ≤ t. Finally,

A ≤ C|Qk|

trivially, since |θt 1 | ≤ C .

2.1. Application: L2 bounds for layer potentials associated to ellip-tic equations in divergence form. Our setting is Rn+1 = (x, t) : x ∈Rn, t ∈ (−∞,∞). Let A = A(x) be a t-independent (n+ 1)× (n+ 1), L∞,elliptic matrix, and let Γ(X,Y ) be the fundamental solution for

L = −divx,t A(x) ∇x,t

in Rn+1, where X = (x, t) and Y = (y, s). By t-independence, Γ(x, t, y, s) =Γ(x, t− s, y, 0).

The single layer potential in Rn+1± is defined as

Stf(x) :=∫

Rn

Γ(x, t, y, 0)f(y)dy

16 STEVE HOFMANN

We are interested in the following estimate:

(2.19) supt‖ ∇x,tStf ‖L2(Rn)≤ C‖f‖L2(Rn) .

2.20. Theorem. [Essentially [AAAHK]] Suppose that the cofficient matrixA has real entries. Let k± and k∗± denote the Poisson kernels for L and L∗

(respectively) in Rn+1± . If there exists a q > 1 such that k±, k∗± ∈ RHq, then

(2.19) holds.

A function k is in the reverse Holder class q, denoted by k ∈ RHq, if forany cube Q (at least locally), one has(∫

Qkq

) 1q

≤ C

∫Qk .

2.21. Remark. Theorem 2.20 applies with q = 2 if A is real and symmetric[JK], and, if n = 1 (i.e., in R2

±), for some q > 1, if A is real, but notnecessarily symmetric [KKPT]. In the latter setting, boundedness of thelayer potentials was established previously by Kenig and Rule [KR].

In the real symmetric case, L2 invertibility of the layer potentials alsoholds, in addition to the boundedness result (2.19). Moreover, boundednessplus invertibility of layer potentials is stable under small complex perturba-tions. All of these results are obtained in [AAAHK], and are related to thelectures of Auscher, Axelsson and McIntosh [AAMc].

Proof. (Sketch of proof of Theorem 2.20)As a preliminary step, we observe that the following estimate holds:

supt>0

‖ ∇xStf ‖22

. supt>0

‖ ∂tStf ‖22 +

∫ ∞

0

∫Rn

|∂2sSsf(x)|2 dx sds+ ‖ f ‖2

2 .

This bound is not obvious, but can be obtained from the solvability of theKato problem for the operator −divx A∗ ∇x, where A∗ is the n × n sub-matrix in the upper left corner of the adjoint matrix A∗, i.e.

A∗i,j = A∗i,j = Aj,i, 1 ≤ i, j ≤ n.

We refer the reader to [AAAHK], Lemma 5.2 for the details.Since k+ ∈ RHq, we have that

‖ ∂tStf ‖22 .

∫ ∞

0

∫Rn

|∇∂sSsf(x)|2 dx sds(2.22)

.∫ ∞

0

∫Rn

|∂2sSsf(x)|2 dx sds ,(2.23)

where (2.22) follows from [DJK], since u = ∂tStf solves Lu = 0 and alsosatisfies that u → 0 at ∞; in (2.23) we have integrated by parts in thevariable s and used Caccioppoli’s inequality.


As a consequence, it is enough to prove∫ ∞

0

∫Rn

|∂2sSsf(x)|2 dx sds ≤ C ‖ f ‖2

2 .

To this end, set θs = s∂2sSs. Its kernel is ψs(x, y) = s∂2

sΓ(x, s, y, 0), which,by the De Giorgi/Nash/Moser estimates, satisfies the standard LP bounds(1.18) and (1.19), as well as (1.30). Hence, it is enough to verify the hy-potheses of the local Tb theorem for square functions (in its Lq version),Theorem 1.26.

Set bQ := |Q|(k∗−)A−Q , where A−Q = (xQ,−`(Q)), and xQ is the center

of Q, and(k∗−)A−Q means the Poisson kernel k∗− with pole at the point A−Q.

Then, by our RHq assumption and normalization of the Poisson kernel,∫Q|bQ|q ≤ C0,

from which the global bound in hypothesis (1.31) may be obtained by wellknown elliptic PDE arguments.

Hypothesis (1.29) follows immediately from the estimate of [CFMS]:∫QbQ =

(ω∗−)A−Q (Q) ≥ δ ,

where(ω∗−)A−Q denotes the harmonic measure ω∗− with pole at A−Q.

Finally, to establish hypothesis (1.32), we first observe that

|s∂2sSsbQ| .

s

`(Q).

Indeed,

s∂2sSsbQ(x) = s|Q|

∫∂2

sΓ(x, s, y, 0) d(ω∗−)A−Q (y)

= s|Q|∂2sΓ(x, s,A−Q)

. s|Q||(x, s)−A−Q|−n−1

.s

`(Q).

Thus, ∫ `(Q)

0

∫Q|s∂2

sSsbQ|2 dxds

s≤ C0 ,

which is the case q = 2 of (1.32); the case 1 < q < 2 follows from the latterbound by Holder’s inequality.

18 STEVE HOFMANN

3. Lecture 3: Local Tb theorem for SIO’s

The following is an extension of M. Christ’s theorem 1.13.

3.1. Local Tb Theorem for Singular Integral Operators. [essentially[AHMTT]; also [AY]] Let T be an SIO with kernel K(x, y) ∈ L∞ (quali-tatively.) Suppose that there exist C0 < ∞, δ > 0, and two systems offunctions b1Q, b2Q, indexed on the dyadic cubes, with biQ, i = 1, 2 sup-ported in Q, such that

(i)∫

Q|biQ|2 ≤ C0, i = 1, 2

(ii)∫

Q|Tb1Q|2 +

∫Q|T

∗b2Q|2 ≤ C0

(iii) δ ≤ |∫

Q biQ|, i = 1, 2.

Then T : L2 → L2.

3.2. Remark. Theorem 3.1 was proved in [AHMTT] for “perfect dyadic”SIOs. More precisely, these are SIOs such that if ψ is supported in a dyadiccube Q, and

∫ψ = 0, then Tψ is supported in Q. The proof extends to the

case of standard SIOs, under the stronger hypothesis∫

Q|biQ|2+ε ≤ C0 (the

details may be found in the unpublished manuscript [H2]). Alternatively,Auscher-Yan [AY] prove a decomposition of a standard SIO into a perfectdyadic SIO + an L2 bounded operator, allowing them to deduce the standardSIO case as a corollary of the perfect dyadic case.

The case of L∞ control is Christ’s Theorem 1.13.Theorem 3.1 extends to the setting of spaces of homogeneous type [A2].

3.1. Application. L2 bounds for layer potentials when the estimate of[DJK] is not available. Indeed, recall that, given the “square function/non-tangential maximal function” estimates of [DJK], we have developed in theprevious section a method for deducing boundedness of layer potentials viathe boundedness of associated square functions, which in turn are handledby Theorem 1.26. In settings where [DJK] type results are unavailable (per-haps simply because the question is open), one might still be able to use someversion of Theorem 3.1 to establish bounds for layer potentials directly. Aparticular example is discussed in the next application.

3.2. Application. Let E be an ADR set (of dimension n), E ⊆ Rn+1, i.e.there exists a constant C0 such that for any x ∈ E, 0 < r < r0,

1C0rn ≤ Hn(B(x, r) ∩ E) ≤ C0r

n ,

where Hn denotes n-dimensional Hausdorff measure.Suppose that E = ∂Ω for some domain Ω ⊆ Rn+1 (e.g., Ω = Rn+1 \ E.)

In particular, E is a space of homogeneous type, and has a “dyadic cube”structure.

The notion of an SIO as defined in 1.1 can be extended to this setting,and harmonic layer potentials are prototypical examples of such SIOs. David


and Semmes characterized the ADR sets E on which all “nice” SIOs are L2

bounded (with respect to the Hn measure), namely the so-called “uniformlyrectifiable” sets.

Let us not impose in advance any hypothesis of uniform rectifiability, butsuppose that for any x ∈ E and 0 < r < r0, we have

(3.3) |B(x, r) ∩ Ω| ≥ 1C1rn+1 .

Notice that (3.3) is of course true if Ω = Rn+1 \ E. Notice also that we donot impose (3.3) in

(Ω)c = Rn+1 \ Ω.

Then by ADR and pigeon-holing, there exists a “corkscrew point” Ax,r ∈Ω such that dist(Ax,r, E) ≈ |Ax,r−x| ≈ r (but notice that we do not imposeany such assumption in Ωc).

We recall that if there is a corkscrew point at all scales r < r0, relativeto every x ∈ E, in both Ω and in Ω− =

(Ω)c, then a theorem of David

and Jerison [DJe] implies that such a ∂Ω is uniformly rectifiable, and henceall SIOs are bounded. Thus, a significant point of our discussion here andbelow is that we impose no such assumption on Ω−.

3.4. Lemma. [Bourgain [B]] For E,Ω as above, given a “dyadic cube”Q ⊆ E, there exists a point AQ with dist(AQ, Q) ≈ dist(AQ, E) ≈ diamQ,such that

ωAQ(Q) ≥ δ > 0 ,

where δ is uniform in Q, and ω is harmonic measure.

We now impose a further condition on ∂Ω, namely that there exists anouter unit normal ν at a.e. x ∈ ∂Ω, and that the Gauss-Green formulaholds in Ω (technically, this amounts to saying that Ω has “locally finiteperimeter”, and that its “measure theoretic boundary” ∂∗Ω coincides with∂Ω a.e. - see [EG], Chapter 5 for the theory of such domains).

3.5. Remark. For such Ω, Wiener’s regularity criterion holds at a.e. pointon ∂Ω (assuming ADR). Also, ADR =⇒ locally finite perimeter.

Next, we present a hitherto unpublished result relating layer potentialsand Poisson kernels in this setting (cf. Remark 3.7 below for some context).

3.6. Proposition. Suppose ωAQ is absolutely continuous with respect toHn|∂Ω, and that

kAQ :=dωAQ

dHn=−∂G(·, AQ)

∂ν∈ L2

loc ,

with ∫eQ∣∣∣kAQ

∣∣∣2dHn ≤ C

|Q|, uniformly in Q ,

20 STEVE HOFMANN

where Q is a fattened version of Q, G is the Green’s function, and we usethe notational convention that for Q ⊂ ∂Ω, |Q| := Hn(Q). Then

∇S : L2(∂Ω) → L2(∂Ω) ,

where

∇Sf(x) “ = ”∫

∂Ω∇xΓ(x− y)f(y) dHn(y),

and

Γ(x) = cn|x|1−n .

3.7. Remark. We do not assume that(Ω)c contains a corkscrew point AQ

for each cube Q ⊆ ∂Ω, nor even that ∂Ω is uniformly rectifiable. On theother hand, it seems reasonable to conjecture that higher integrability of thePoisson kernel implies uniform rectifiability of the boundary, given the sortof background hypotheses that we have imposed here, and it seems likelythat Proposition 3.6 (or some variant of it) may play a role in establishingsuch a conjecture.

Proof. (Sketch of proof of Proposition 3.6) The sketch we present is formal.Making it rigorous seems to require truncations of the kernel, and this ismessy.

We apply the local Tb theorem 3.1 with

bQ := |Q|kAQ 1Q .

Observe that K(x, y) = ∇xΓ(x− y) is a standard, antisymmetric Calderon-Zygmund kernel. Let us verify the hypotheses of the local Tb theorem 3.1.Hypothesis (iii), namely δ ≤

∫Q bQ is just Bourgain’s Proposition 3.4.

Hypothesis (i), namely∫

Q|bQ|2 ≤ C0, holds by assumption.

To establish hypothesis (ii), namely∫Q|TbQ|2 ≤ C0 ,

in a rigorous way, would involve truncating the singular kernels. Instead,suppose x ∈ Ω, and consider

∇xSbQ(x) ,

for x ∈ Ω “near” Q ⊆ ∂Ω. Observe that for such x, we have that

(3.8) |∇xΓ(x−AQ)| ≤ C

|x−AQ|n≤ C

(diamQ)n ≈1|Q|

.

Also, if Q is a “fattened” version of Q, then for x near Q, we have∣∣∣∫ ∇Γ(x− y) 1( eQ)c(y) dωAQ(y)∣∣∣ . 1

(diamQ)n ≈1|Q|

.


Ignoring the fact that

bQ = |Q|kAQ 1Q 6= |Q|kAQ 1 eQ ,

we have (morally) reduced matters to considering∫∇xΓ(x− y) dωAQ(y) .

By (3.8), it is enough to treat the error

∇xΓ(x−AQ)−∇x

∫Γ(x− y) dωAQ(y) = ∇xG(x,AQ) .

If δ(x) := dist(x, ∂Ω), and ∆x := ∂Ω∩B(x,Rδ(x)) for some suitable constantR (depending on the implicit constants in Bourgain’s Lemma), then∣∣∣∇xG(x,AQ)

∣∣∣ . G(x,AQ)δ(x)

(3.9)

.∫

∆x

kAQ . M(kAQ 1 eQ) ,(3.10)

where in (3.9) we have used interior estimates, and in (3.10) we have usedBourgain’s Lemma 3.4 and the maximum principle, as in, e.g., [K], p. 9.

Finally, apply now that, by hypothesis,∫eQ∣∣∣kAQ

∣∣∣2 .1|Q|

.

Proof. (Some ideas of the proof of local Tb theorem for SIOs 3.1)As in the proof of the local Tb theorem for square functions 1.26, we shall

use the local Tb hypotheses to verify the hypotheses of the T1 Theorem (inthis case, that of David and Journe [DJ]). To be precise, we seek to establisha local T1 condition:

(3.11) supQ

∫Q

∣∣∣T 1Q

∣∣∣ ≤ C0 ,

which implies both that T 1 ∈ BMO and WBP (1.6) (we would also needthe corresponding bound for T ∗ 1Q.) In the present heuristic exposition, tosimplify matters, we shall concentrate on the global conditions T 1 ∈ BMOand T ∗ 1 ∈ BMO, rather than (3.11) . We shall ignore the issue of WBP.

Let

∆tf(x) :=∫t−nψ

(x− y

t

)f(y) dy ,

22 STEVE HOFMANN

where ψ ∈ C∞0 (B(0, 1)),

∫ψ = 0, and

∫|ψ(tξ)|2 dt

t = 1. We then have thefollowing “Calderon reproducing formula”∫ ∞

0∆2

t

dt

t= Id ,

with convergence in the strong operator topology in L2. Our goal is to showthat ∣∣∣∆tT 1

∣∣∣2 dxdtt

is a Carleson measure. To this end, we would like to apply the local Tbtheory for square functions to

θt = ∆tT .

However, there is a difficulty here, in that ∆tT does not have a standardLP kernel in general (the kernel is bad “near the diagonal”). On the otherhand, we could overcome this difficulty if T ∗ 1 = 0 (or even if T ∗ 1 ∈ BMO).Indeed, the kernel of ∆tT is (at least formally)

∫ψt(x − z)K(z, y) dz so if

T ∗1 = 0 (formally,∫K(z, y)dz = 0), then the smoothness of ψt can be used

to weaken the singularity of K when |x − y| . t (in practice, this is a bitdelicate, and seems to require that one assume WBP).

As in [DJS] and [CJS] (although, to the present author’s knowledge, theidea seems to have originated with Coifman), we deal with this difficultyby building ∆t adapted to b2Q (in our case, this is a local construction ofcourse), since T ∗b2Q is good locally on Q (maybe not = 0, but still ok.) I.e.the control on T ∗b2Q substitutes for T ∗ 1 = 0, and allows one (eventually) tohandle

∫ψt(x, z) b2Q(z) K(z, y) dz.

It is easier to carry out this strategy in a discretized setting. Supposeb is accretive. Let Dk denote the grid of dyadic cubes with `(Q) = 2−k.Let Ekf denote the dyadic averaging operator at scale 2−k, i.e. Ekf =∑

Q∈Dk1Q

∫Q f , and define the corresponding martingale difference opera-

tors ∆k = Ek+1 − Ek. Notice that ∆k1 = 0. One can build a discretizedLittlewood-Paley theory with the ∆k.

Now consider an adapted version of Ek and ∆k (following [CJS]). Moreprecisely, given an accretive b, set

Ebkf =

Ek(bf)Ek(b)

,

and notice that Ebk1 = 1. Consider the associated martingale differences

∆bk = Eb

k+1 − Ebk .


Then ∆bk kills constants and, for an accretive b, there is a nice Littlewood-

Paley theory for ∆bk, i.e.∫

Rn

∑k

|∆bkf |2 dx .‖ f ‖2

2 .

Now, in lieu of the continuous Carleson region RQ as above, we considera discrete analogue, ∫

Q

∑k:2−k≤`(Q)

∣∣∣∆b2Qk T 1

∣∣∣2 dx .Here we face another difficulty, namely, that E

b2Qk f =

Ek(b2Qf)

Ek(b2Q)is a good

expression only where b2Q is accretive. We know that b2Q has a “big” averageover its own cube Q, i.e., that |

∫Q b

2Q| > δ > 0 , but it is conceivable that

the expectations over the smaller subcubes of Q could be small, i.e. Ek(b2Q)could be small.

In order to fix this problem, we first need to extract, via a stopping timeargument, an ample sawtooth on which∣∣∣Ek(b2Q)

∣∣∣ ≥ δ ,

because in that sawtooth, the ∆b2Qk are good operators.

Working with ∆b2Qk (T 1) allows us to exploit the good behaviour of T ∗b2Q

on Q, so that the kernel of θk = ∆b2Qk T is “close” to a standard LP kernel,

and we can proceed at this point more or less as in the proof of the local Tbtheorem for square functions with respect to b1Q (i.e. the system of functionsfor T , not for T ∗.) That is, by another stopping time for b1Q, we build a“sawtooth inside a sawtooth”, in which we have∣∣∣∆b2Q

k T 1∣∣∣ ≤ ∣∣∣(∆

b2Qk T 1

)Ekb

1Q

∣∣∣ .We then proceed roughly as in the proof of Theorem 1.26. In practice thisis all rather delicate, and we refer the reader to [AHMTT] for details.

3.3. Open Problems.

3.3.1. Local Tb with Lq control. Prove a local Tb theorem for standardCalderon-Zygmund kernels, with Lq control, with q > 1, i.e. with∫

Q|bQ|q ≤ C0 .

24 STEVE HOFMANN

We have seen that this works for square functions. Also, the argument of[AHMTT] works for perfect dyadic SIOs (but with Tb1Q , T ∗b2Q ∈ Lq′

if b1Q, b2Q ∈ Lq, where, as usual, 1

q + 1q′ = 1.) However, the error terms

(the Calderon-Zygmund tails) that arise in the case of standard SIOs seemintractable if q < 2. The method of [AY], by which the result for standardSIOs is deduced from that for perfect dyadic operators, seems inapplicablehere, unless one could prove a stronger version of the perfect dyadic versionrequiring only Lq (not Lq′) control of Tb1Q and T ∗b2Q.

A result of this type is likely to be useful in applications to layer potentials,and to free boundary theory (cf. Remark 3.7).

3.3.2. Matrix-valued local Tb for SIOs. Prove a matrix-valued version of thelocal Tb theorem for SIOs (i.e. for BQ that are matrix-valued.) We haveseen that this works for square functions (in particular, for the solution ofthe Kato problem). The difficulty for SIOs lies in the fact that when usingadapted expectation operators, we would presumably need to consider

EBQ

k~f = (EkBQ)−1Ek

(BQ

~f).

How then do we extract an ample sawtooth on which EkBQ is invertible(with uniform control on the inverse (EkBQ)−1)?

One possible approach might be to revisit the ideas of the first proof of the2-dimensional Kato problem [HMc], in which the stopping time procedureof Theorems 1.26 and 1.35 was carried out with respect to determinants ofmatrixes (this would require not L2 but Ln+ε integrability); that argumentused in an apparently crucial way that the appropriate matrix in that contextwas not arbitrary, but rather the derivative of a mapping. On the otherhand, perhaps one can figure out a way to directionalize as in the proof ofTheorem 1.35 (in the proof of the latter Theorem, and in the solution of theKato problem, one could control a given direction at a time, but one doesnot produce a sawtooth good for all directions simultaneously).

We suspect the solution to this problem might be applicable to the de-velopment of the layer potential method for strongly elliptic systems withvariable coefficients.

3.3.3. Connection between Poisson kernel bounds and layer potentials. Re-call that we have observed that Lq estimates for Poisson kernels can beused to prove boundedness of layer potentials (bQ = |Q|kAQ , or perhapsbQ = |Q|kAQ 1Q ). On the other hand it is known by abstract functionalanalysis that, given an L2 bounded SIO T , there exist pseudo-accretive sys-tems b1Q, b2Q, adapted to T and T ∗ with L∞ control.

Problem: Can we make this connection explicit in the case of layerpotentials and Poisson kernels; i.e. if layer potentials are bounded on L2,does this imply some sort of non-degeneracy of the Poisson kernel k (e.g.


log k ∈ BMO, at least in some “big pieces” sense...)? If this were the case,it would have very significant applications.

There is indirect evidence for this conjecture in 2 dimensions: L2 bound-edness of the Cauchy transform on an ADR set implies that the set is uni-formly rectifiable [MMV], and, in turn, for simply connected domains inR2 with ∂Ω ADR and uniformly rectifiable, there is a non-degeneracy forharmonic measure ω [BiJo].

3.3.4. Boundedness of variable coefficient layer potentials. Let

L = −divx,t A(x) ∇x,t

in Rn+1, where A is an (n + 1) × (n + 1), t-independent, complex, elliptic(accretive), L∞ matrix.

Conjecture: The associated layer potentials are bounded on L2.By [AAAHK], the set of coefficient matrices for which we have (simultane-

ously) boundedness, invertibility and square function bounds is open. Thus,in particular, the conjecture has been solved in a complex neighborhood ofany real, symmetric matrix (as above).

The block case where the matrix A consists of an n × n block B matrixin the upper left corner (i.e. Ai,j = Bi,j for 1 ≤ i, j ≤ n), and An+1,n+1 = 1and otherwise Ai,j = 0 (i.e. Ai,n+1 = 0 = An+1,i for 1 ≤ i ≤ n), is the Katoproblem.

Question: Is the work of Auscher-Axelsson-McIntosh [AAMc] applicablehere? In the block case, yes, because the result of [AAMc] includes thesolution of the Kato problem as a special case.

Acknowledgment. These lectures are an expanded and updated version ofmy ICM lecture of the same title [H], and were presented at the meeting onHarmonic Analysis and PDE held at El Escorial in June 2008. I thank theorganizers of the meeting for their gracious hospitality, and for providingme with this opportunity. I am also indebted to Ignacio Uriarte-Tuero, forhis invaluable help in typesetting my handwritten lecture notes. Finally,I thank the referee for several useful suggestions which have improved theexposition of the paper.

References

[AAAHK] M. Alfonseca, P. Auscher, A. Axelsson, S. Hofmann and S. Kim, Stability oflayer potentials and L2 solvability of boundary value problems for divergence formelliptic equations with complex L∞ coefficients, preprint.

[A] P. Auscher, Lectures on the Kato square root problem, Surveys in analysis andoperator theory (Canberra, 2001), 1–18, Proc. Centre Math. Appl. Austral. Nat.Univ. 40, Austral. Nat. Univ., Canberra, 2002.

[A2] P. Auscher, personal communication.[AAMc] P. Auscher, A. Axelsson, and A. McIntosh, On a quadratic estimate related to

the Kato conjecture and boundary value problems, these proceedings.[AHLMcT] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and P. Tchamitchian, The

solution of the Kato Square Root Problem for Second Order Elliptic operators onRn, Annals of Math. 156 (2002), 633–654.

26 STEVE HOFMANN

[AHLT] P. Auscher, S. Hofmann, J.L. Lewis, P. Tchamitchian, Extrapolation of Carlesonmeasures and the analyticity of Kato’s square root operators, Acta Math. 187(2001), no. 2, 161–190.

[AHMTT] P. Auscher, S. Hofmann, C. Muscalu, T. Tao, C. Thiele, Carleson measures,trees, extrapolation, and T (b) theorems, Publ. Mat., 46 (2002), no. 2, 257–325.

[AY] P. Auscher and Q. X. Yang, On local T (b) Theorems, Pub. Math. 53 (2009), 179-196.

[AT] P. Auscher and Ph. Tchamitchian, Square root problem for divergence operatorsand related topics, Asterisque Vol. 249 (1998), Societe Mathematique de France.

[BiJo] C. Bishop and P. Jones, Harmonic Measure and arclength, Annals of Math. 132(1990), 511-547.

[B] J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimen-sions, Invent. Math. 87, 477-483.

[CFMS] L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegativesolutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981),no. 4, 621–640.

[C1] A. P. Calderon, Commutators of singular integral operators, Proc. Nat. Acad. Sci.USA, 53, 1965, 1092-1099.

[C2] A. P. Calderon, Commutators, singular integrals on Lipschitz curves, and applica-tions, Proceedings of the ICM Helsinki 1978, Acad. Sci. Fennica, Helsinki, 1980,85-96.

[Ch] M. Christ, A T(b) theorem with remarks on analytic capacity and the Cauchyintegral, Colloquium Mathematicum LX/LXI (1990) 601-628.

[CJ] M. Christ and J.-L. Journe, Polynomial growth estimates for multilinear singularintegral operators. Acta Math. 159 (1987), no. 1-2, 51–80.

[CJS] R. Coifman, P. Jones and S. Semmes, Two elementary proofs of the L2 boundednessof Cauchy integrals on Lipschitz curves. J. Amer. Math. Soc. 2 (1989), no. 3, 553–564.

[CMcM] R. Coifman, A. McIntosh, and Y. Meyer. L’integrale de Cauchy definit unoperateur borne sur L2 pour les courbes lipschitziennes. Annals of Math. 116 361–387, 1982.

[CM] R. Coifman and Y. Meyer. Non-linear harmonic analysis and PDE, in E. M. Stein,editor, Beijing Lectures in Harmonic Analysis, volume 112 of Annals of Math.Studies. Princeton Univ. Press, 1986.

[DJK] B. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differen-tial operators with nonsmooth coefficients, Ark. Mat., 22 (1984), no. 1, 97–108.

[D] G. David, Unrectifiable 1-sets have vanishing analytic capacity. Rev. Mat.Iberoamericana 14 (1998), no. 2, 369–479.

[DJe] G.David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic mea-sure, and singular integrals, Indiana Univ. Math. J., 39 (1990), no. 3, 831–845.

[DJ] G. David and J.-L. Journe, A boundedness criterion for generalized Calderon-Zygmund operators. Ann. Math. 120 371–398, 1984.

[DJS] G. David, J.-L. Journe, and S. Semmes, Operateurs de Calderon-Zygmund, fonc-tions para-accretives et interpolation. Rev. Mat. Iberoamericana 1 1–56, 1985.

[EG] L.C. Evans and R.F.Gariepy, Measure Theory and Fine Properties of Functions,Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[FS] C. Fefferman, and E. M. Stein, Hp spaces of several variables. Acta Math. 129(1972), no. 3-4, 137–193.

[H] S. Hofmann, Local Tb Theorems and applications in PDE, Proceedings of the ICMMadrid 2006.

[H2] S. Hofmann, A proof of the local Tb Theorem for standardCalderon-Zygmund operators, unpublished manuscript posted athttp://www.math.missouri.edu/˜hofmann/


[HLMc] S. Hofmann, M. Lacey and A. McIntosh. The solution of the Kato problem fordivergence form elliptic operators with Gaussian heat kernel bounds. Annals ofMath. 156 (2002), pp 623-631.

[HMc] S. Hofmann and A. McIntosh, The solution of the Kato problem in two dimensions,Proceedings of the Conference on Harmonic Analysis and PDE held in El Escorial,Spain in July 2000, Publ. Mat. Vol. extra, 2002 pp. 143-160.

[JK] D. Jerison and C. Kenig, The Dirichlet problem in nonsmooth domains, Ann. ofMath. (2) 113 (1981), no. 2, 367–382.

[K] C. Kenig, Harmonic analysis techniques for second order elliptic boundary valueproblems, CBMS Regional Conference Series in Mathematics, 83. Published for theConference Board of the Mathematical Sciences, Washington, DC; by the AmericanMathematical Society, Providence, RI, 1994

[KKPT] C. Kenig, H. Koch, H. J. Pipher and T. Toro, A new approach to absolutecontinuity of elliptic measure, with applications to non-symmetric equations, Adv.Math., 153 (2000), no. 2, 231–298.

[KR] C. Kenig and D. Rule, The regularity and Neumann problems for non-symmetricelliptic operators, Trans. Amer. Math. Soc. 361 (2009), 125-160.

[McM] A. McIntosh and Y. Meyer. Algebres d’operateurs definis par des integrales sin-gulieres. C. R. Acad. Sci. Paris 301 Serie 1 395–397, 1985.

[MMV] P. Mattila, M. Melnikov and J. Verdera, The Cauchy integral, analytic capacity,and uniform rectifiability. Ann. of Math. (2) 144 (1996), no. 1, 127–136.

[NTV1] F. Nazarov, S. Treil and A. Volberg, Accretive system Tb-theorems on nonhomo-geneous spaces. Duke Math. J. 113 (2002), no. 2, 259–312.

[NTV2] F. Nazarov, S. Treil and A. Volberg, The Tb-theorem on non-homogeneous spaces.Acta Math. 190 (2003), no. 2, 151–239.

[P] J. Peetre, On convolution operators leaving Lp, λ spaces invariant, Ann. Mat. PuraAppl. (4) 72 (1966) 295–304.

[S] S. Semmes. Square function estimates and the T (b) Theorem. Proc. Amer. Math.Soc. 110 (1990), no. 3, 721–726.

[Sp] S. Spanne, Sur l’interpolation entre les espaces LkpΦ, Ann. Scuola Norm. Sup. Pisa

(3) 20 (1966) 625–648.[St] E. M. Stein, Singular integrals, harmonic functions, and differentiability proper-

ties of functions of several variables, Singular integrals Proc. Sympos. Pure Math.,Chicago, Ill., 1966, pp. 316–335, Amer. Math. Soc., Providence, R.I., 1967.

[T] X. Tolsa, Painleve’s problem and the semiadditivity of analytic capacity. Acta Math.190 (2003), no. 1, 105–149.

[Vo] A. Volberg, Caldern-Zygmund capacities and operators on nonhomogeneous spaces,CBMS Regional Conference Series in Mathematics 100. Published for the Con-ference Board of the Mathematical Sciences, Washington, DC; by the AmericanMathematical Society, Providence, RI, 2003.

Mathematics Department, University of Missouri, Columbia, MO 65211 USA

LOCAL T THEOREMS AND APPLICATIONS IN...

Documents

Transcript of LOCAL T THEOREMS AND APPLICATIONS IN...