Classi cation of Poincar e inequalities and PI-recti...

Classification of Poincare inequalities and PI-rectifiablity

Classification of Poincare inequalities andPI-rectifiablity

Sylvester Eriksson–Bique

Courant Institute – New York University (Soon: NYU)Warick University GMT Workshop

July 14th 2017


Standing assumption

(X , d , µ) proper metric measure space, µ Radon measure.

Lip f (x) = lim supx 6=y→x

|f (x)− f (y)|d(x , y)


Poincare inequality

For every Lipschitz f : X → R

B(x ,r)

|f − fB(x ,r)| dµ ≤ Cr

( B(x ,C ′r)

Lip f p dµ

) 1p

. (1)

Definition

(X , d , µ) is a ((1, p)-)PI-space if µ is doubling and the spacesatisfies a ((1, p))-Poincare inequality.

Name Dropping: Heinonen,Koskela, Keith, Zhong,Shanmugalingam, Laakso, Maly, Korte, Dejarnette, J. Bjorn,Kleiner, Cheeger, Schioppa


Quote

From Heinonen (’05, published ’07, based on talk in ’03):

“How does one recognize doubling p-Poincare spaces? Do suchspaces, apart from certain trivial or standard examples, occurnaturally in mathematics? The answer to the second question is aresounding yes...The answer to the first question is morecomplicated. There exist techniques that can be employed here;some are similar to those which we used earlier to prove that aPoincare inequality holds in Rn. On the other hand, most of thecurrently known techniques are quite ad hoc, and there is room forimprovement.”


Main questions

Which conditions characterize PI-spaces?

How does the exponent p depend on the geometry of thespace?

Relationships to differentiability spaces?


Classical view on Poincare

In terms of Modulus of some family Γ, with respect to ameasure ν,

infρ

ˆBρp dµ,

where ρ admissible, i.e.´γ ρ ≥ 1 for all γ ∈ Γ.

Poincare inequality related to lower bounds for modulus.


Prior characterization and downside

Several and in different contexts: Heinonen-Koskela, Keith,Shanmugalingam-Jaramillo-Durand-Caragena, Bonk-Kleiner

Downsides: Usually requires curve family to estimate relevantmodulus, regularity or knowledge of p.

Not ideal for studying abstract differentiability spaces, sinceonly weaker conditions can be obtained directly.


Obligatory Slide

Theorem (Rademacher’s theorem)

Every Lipschitz f : Rn → R is differentiable almost everywhere.

Theorem (Cheeger ’99, Metric Rademacher’s Theorem)

Every PI-space is a Lipschitz Differentiability space (LDS), i.e.every Lipschitz function is almost every where differentiable tosome given charts.


Measurable differentiable structure for (X , d , µ)

Measurable sets Ui , Lip-functions φi : X → Rni

µ(X \⋃Ui ) = 0

Every Lip function f : X → RN , for every i and almost everyx ∈ Ui has a unique derivative dfi (x) : Rni → RN s.t.

f (y)− f (x) = dfi (x)(φi (y)− φi (x)) + o(d(x , y)).

If such a structure exists, (X , d , µ) is a LDS.

Introduced by Cheeger, axiomatized by Keith.


Again, from Heinonen:

“An important open problem is to understand what exactly isneeded for the conclusions in Cheegers work.”


More precise question

Question

Are the assumptions of Cheeger (PI and doubling) necessary? Doesa differentiability space have a Poincare inequality, in some form?

May be totally disconnected! E.g. fat Cantor set

Need to be careful about how to phrase a question


Even more precise question

Question

Are differentiability spaces PI-rectifiable, that is can everydifferentiability space be covered up to a null-set by positivemeasure isometric subsets of PI-spaces?

Stated formally by Cheeger, Kleiner and Schioppa. Answer: NO

Theorem (Schioppa 2016)

A construction of (X , d , µ) which is LDS, but not PI-rectifiable.


RNP-Measurable differentiable structure for (X , d , µ)

Measurable sets Ui , Lip-functions φi : X → Rni

µ(X \⋃Ui ) = 0

V is an arbitrary RNP-Banach space (Lp, lp, c0, NOT L1)

Every Lip function f : X → V , for every i and almost everyx ∈ Ui has a unique derivative dfi (x) : Rni → V s.t.

f (y)− f (x) = dfi (x)(φi (y)− φi (x)) + o(d(x , y)).

If such a structure exists, (X , d , µ) is a RNP-LDS(RNP-Lipschitz Differentiability Space)

Used by Cheeger and Kleiner, defined/studied by Bate and Li


Cheeger-Kleiner

Theorem (Cheeger-Kleiner)

Every PI-space is a RNP-LDS.


Positive result

Theorem (Bate, Li 2015)

If (X , d , µ) is a RNP-LDS, then at almost every point“Alberti-representations connect points” (asymptotic connectivity).[Also: Asymptotic non-hoomogeneous Poincare.]

Theorem (E-B, 2016)

A proper metric measure space (X , d , µ) equipped with a Radonmeasure µ is a RNP-Lipschitz differentiability space if and only if itis PI-rectifiable (and all σ-porous sets have zero measure).

Corollary: Andrea Schioppa’s example is not RNP-Lipschitzdifferentiability. (Could be also obtained directly.)


Proof: Problems in proving rectifiability

How to identify a decomposition to good pieces Ui? (Bateand Li already identified these, and used them to proveweaker PI-type results). Has doubling and connectivityproperties “relative to X”.

Enlarge these Ui to “connected” metric spaces Ui by glueinga “tree-like” graph to it, which approximates a neighborhoodin X .

How to establish Poincare inequalities for Ui usingdifferentiability? Which exponent p? Characterizing PI usingconnectivity.

Subsets a priori disconnected


Definition (E-B ’16, motivated by similar conitions in Bate-Li ’15)

1 < C , 0 < δ, ε < 1 given

X is (C , δ, ε)-connected

If for every x , y ∈ X , d(x , y) = r ,

and every obstacle E (x , y 6∈ E ) with

µ(E ∩ B(x ,Cr)) < εµ(B(x ,Cr)),

there exists a 1-Lip curve fragment γ : K → X almostavoiding E , i.e.

1 γ(max(K )) = y , γ(min(K )) = x2 max(K )−min(K ) ≤ Cr3 γ(K ) ∩ E = ∅4 |[min(K ),max(K )] \ K | ≤ δr


Improving the estimate

(C , δ, ε)-connected for some 0 < δ, ε < 1, implies(C ′,C ′′τα, τ)-connectivity for some 0 < α < 1 and all 0 < τ .

Note, Li-Bate obtained (C ,C ′g(τ), τ)-asymptotic connectivityfor some g going to zero, but no quantitative control: we useiteration to obtain the polynomial control for g .

Once α is identified, 1/p-Poincare holds for p > 1α .

Crucial idea: Maximal function estimate, and re-applying theestimate to the gaps.


Main Theorem

Theorem

(E-B 2016) A (D, r0)-doubling (X , d , µ) is (C , δ, ε)-connected forsome 0 < δ, ε < 1 iff it is (1, q)-PI for some q > 1 (possibly large).

Connectivity can be established in many cases naturally, withoutknowing p!


Back to PI-rectifiability: Thickening


Theorem (E-B, 2016)

A proper metric measure space (X , d , µ) equipped with a Radonmeasure µ is a RNP-Lipschitz differentiability space if and only if itis PI-rectifiable (and all σ-porous sets have zero measure).


Starting point

If (X , d , µ) (intrinsically) (C , δ, ε)-connected, then PI.

Bate-Li provide subsets Ui ⊂ X , which are “relatively”doubling and “relatively” (and locally) (C , δ, ε)-connected.

Need a way to find something to glue to Ui to get(C ′, δ′, ε′)-connectivity of a larger space Ui , from which thePI-rectifiability follows.


Thickening Lemma (E-B 2016)

Main tool in proving rectifiability result.Let r0 > 0 be arbitrary.

(X , d , µ) proper metric measure, and K ⊂ X compact,

X doubling (simplifying assumption), and

Pairs (x , y) ∈ K are (C , δ, ε)-connected in X


Then:

There exists constants C , ε,D > 0

A complete metric space K which is D-doubling and“well”-connected

An isometry ι : K → K which preserves the measure.

The resulting metric measure space K is a PI-space.


Hyperbolic filling

Choose centers p ∈ Ni (2−i -nets of K ), and p ∈Wi (Whitneycenters at level 2−i in the complement), r(p) = 2−i

Define a graph by declaring pairs v = (p, r(p)) to be vertices

Connect v = (p, r(p)) and w = (q, r(q)) if

1 d(p, q) ≤ C (r(p) + r(q))2 1/2 ≤ r(p)/r(q) ≤ 2


Metric measure graph

Declare the edge to have length C ′(r(p) + r(q))

and associate measure proportional toµ(B(p, r(p))) + µ(B(q, r(q))).

Doubling correspnds to doubling of X , and curves can beapproximated by curves in the graph.

Connectivity of X along the subset corresponds to connectivityof the graph. Need a connectivity condition involving sets.

Similar to some arguments by Kleiner-Bonk onquasisymmetric parametrizations of spheres.


Relationship to Muckenhoupt weights

Introduced by Muckenhoupt. “Quantitative absolutecontinuity. w ∈ A∞ if one of the following

1 For some 0 < δ < 1 there exists a 0 < ε < 1 such that for anyball B ⊂ Rn and any E ⊂ Q

w(E ) ≤ εw(B) =⇒ λ(E ) ≤ δw(B)

2 There exists an 0 < α < 1 such that

λ(E )

λ(B)≤ C

(w(E )

w(B)

)α


PI-version

(X , d , µ) is PI, if it is D-doubling and for every

1 For some 0 < δ < 1 (sufficiently small) there exists a0 < ε < 1 such tha for any x , y ∈ X and every E ⊂ B(x ,Cr)

µ(E ) ≤ εµ(CB) =⇒

There is a curve γ connecting the pair such that

H1|γ(E ) ≤ δH1|γ(CB)

2 There exists an 0 < α < 1 such that for every set E thereexists a curve γ connecting the pair of points such that

H1|γ(E )

H1|γ(CB)≤ C

(µ(E )

µ(CB)

)α


Application

Definition

limt→0 Φ(t) = Φ(0) = 0

limt→0 Ψ(t) = Ψ(0) = 0

If for every 2-Lipschitz f and every B(x , r)

B(x ,r)

|f − fB(x ,r)| dµ ≤ rΨ

( B(x ,Cr)

Φ ◦ Lip f dµ

).

then (X , d , µ) satisfies (Φ,Ψ,C )-non-homogeneous-Poincare.(NHP)


Application

Theorem (E-B (2016))

If a D-doubling (X , d , µ) satisfies a (Φ,Ψ,C )-NHP, then it is aPI-space. I.e. It satisfies (1, q)-Poincare inequality for some,possibly very large q.


Thank you!

Sylvester Eriksson-Bique

[email protected]

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