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Classification of Poincar´ e inequalities and PI-rectifiablity

Classification of Poincar´ e inequalities and PI-rectifiablity

Sylvester Eriksson–Bique

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

July 14th 2017

Classification of Poincar´ e inequalities and PI-rectifiablity

Standing assumption

(X, d, µ) proper metric measure space, µ Radon measure. Lip f (x) = lim sup

x=y→x

|f (x) − f (y)| d(x, y)

Classification of Poincar´ e inequalities and PI-rectifiablity

Poincar´ e inequality

For every Lipschitz f : X → R

B(x,r)

|f − fB(x,r)| dµ ≤ Cr

• B(x,C ′r)

Lip f p dµ 1

p

. (1) Definition (X, d, µ) is a ((1, p)-)PI-space if µ is doubling and the space satisfies a ((1, p))-Poincar´ e inequality. Name Dropping: Heinonen,Koskela, Keith, Zhong, Shanmugalingam, Laakso, Maly, Korte, Dejarnette, J. Bj¨

• rn,

Kleiner, Cheeger, Schioppa

Classification of Poincar´ e inequalities and PI-rectifiablity

Quote

From Heinonen (’05, published ’07, based on talk in ’03): “How does one recognize doubling p-Poincar´ e spaces? Do such spaces, apart from certain trivial or standard examples, occur naturally in mathematics? The answer to the second question is a resounding yes...The answer to the first question is more

• complicated. There exist techniques that can be employed here;

some are similar to those which we used earlier to prove that a Poincare inequality holds in Rn. On the other hand, most of the currently known techniques are quite ad hoc, and there is room for improvement.”

Classification of Poincar´ e inequalities and PI-rectifiablity

Main questions

Which conditions characterize PI-spaces? How does the exponent p depend on the geometry of the space? Relationships to differentiability spaces?

Classification of Poincar´ e inequalities and PI-rectifiablity

Classical view on Poincar´ e

In terms of Modulus of some family Γ, with respect to a measure ν, inf

ρ

ˆ

B

ρp dµ, where ρ admissible, i.e. ´

γ ρ ≥ 1 for all γ ∈ Γ.

Poincar´ e inequality related to lower bounds for modulus.

Classification of Poincar´ e inequalities and PI-rectifiablity

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 relevant modulus, regularity or knowledge of p. Not ideal for studying abstract differentiability spaces, since

• nly weaker conditions can be obtained directly.

Classification of Poincar´ e inequalities and PI-rectifiablity

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 to some given charts.

Classification of Poincar´ e inequalities and PI-rectifiablity

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 every x ∈ 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.

Classification of Poincar´ e inequalities and PI-rectifiablity

Again, from Heinonen: “An important open problem is to understand what exactly is needed for the conclusions in Cheegers work.”

Classification of Poincar´ e inequalities and PI-rectifiablity

More precise question

Question Are the assumptions of Cheeger (PI and doubling) necessary? Does a differentiability space have a Poincar´ e inequality, in some form? May be totally disconnected! E.g. fat Cantor set Need to be careful about how to phrase a question

Classification of Poincar´ e inequalities and PI-rectifiablity

Even more precise question

Question Are differentiability spaces PI-rectifiable, that is can every differentiability space be covered up to a null-set by positive measure 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.

Classification of Poincar´ e inequalities and PI-rectifiablity

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 every x ∈ 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

Classification of Poincar´ e inequalities and PI-rectifiablity

Cheeger-Kleiner

Theorem (Cheeger-Kleiner) Every PI-space is a RNP-LDS.

Classification of Poincar´ e inequalities and PI-rectifiablity

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 Poincar´ e.] Theorem (E-B, 2016) A proper metric measure space (X, d, µ) equipped with a Radon measure µ is a RNP-Lipschitz differentiability space if and only if it is PI-rectifiable (and all σ-porous sets have zero measure). Corollary: Andrea Schioppa’s example is not RNP-Lipschitz

• differentiability. (Could be also obtained directly.)

Classification of Poincar´ e inequalities and PI-rectifiablity

Proof: Problems in proving rectifiability

How to identify a decomposition to good pieces Ui? (Bate and Li already identified these, and used them to prove weaker PI-type results). Has doubling and connectivity properties “relative to X”. Enlarge these Ui to “connected” metric spaces Ui by glueing a “tree-like” graph to it, which approximates a neighborhood in X. How to establish Poincar´ e inequalities for Ui using differentiability? Which exponent p? Characterizing PI using connectivity. Subsets a priori disconnected

Classification of Poincar´ e inequalities and PI-rectifiablity

Definition (E-B ’16, motivated by similar conitions in Bate-Li ’15) 1 < C, 0 < δ, ǫ < 1 given X is (C, δ, ǫ)-connected If for every x, y ∈ X, d(x, y) = r, and every obstacle E (x, y ∈ E) with µ(E ∩ B(x, Cr)) < ǫµ(B(x, Cr)), there exists a 1-Lip curve fragment γ : K → X almost avoiding E, i.e.

1 γ(max(K)) = y, γ(min(K)) = x 2 max(K) − min(K) ≤ Cr 3 γ(K) ∩ E = ∅ 4 |[min(K), max(K)] \ K| ≤ δr

Classification of Poincar´ e inequalities and PI-rectifiablity

Improving the estimate

(C, δ, ǫ)-connected for some 0 < δ, ǫ < 1, implies (C ′, C ′′τ α, τ)-connectivity for some 0 < α < 1 and all 0 < τ. Note, Li-Bate obtained (C, C ′g(τ), τ)-asymptotic connectivity for some g going to zero, but no quantitative control: we use iteration to obtain the polynomial control for g. Once α is identified, 1/p-Poincar´ e holds for p > 1

α.

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

Classification of Poincar´ e inequalities and PI-rectifiablity

Main Theorem

Theorem (E-B 2016) A (D, r0)-doubling (X, d, µ) is (C, δ, ǫ)-connected for some 0 < δ, ǫ < 1 iff it is (1, q)-PI for some q > 1 (possibly large). Connectivity can be established in many cases naturally, without knowing p!

Classification of Poincar´ e inequalities and PI-rectifiablity

Back to PI-rectifiability: Thickening

Classification of Poincar´ e inequalities and PI-rectifiablity

Theorem (E-B, 2016) A proper metric measure space (X, d, µ) equipped with a Radon measure µ is a RNP-Lipschitz differentiability space if and only if it is PI-rectifiable (and all σ-porous sets have zero measure).

Classification of Poincar´ e inequalities and PI-rectifiablity

Starting point

If (X, d, µ) (intrinsically) (C, δ, ǫ)-connected, then PI. Bate-Li provide subsets Ui ⊂ X, which are “relatively” doubling and “relatively” (and locally) (C, δ, ǫ)-connected. Need a way to find something to glue to Ui to get (C ′, δ′, ǫ′)-connectivity of a larger space Ui, from which the PI-rectifiability follows.

Classification of Poincar´ e inequalities and PI-rectifiablity

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, ǫ)-connected in X

Classification of Poincar´ e inequalities and PI-rectifiablity

Then: There exists constants C, ǫ, 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.

Classification of Poincar´ e inequalities and PI-rectifiablity

Hyperbolic filling

Choose centers p ∈ Ni (2−i-nets of K), and p ∈ Wi (Whitney centers 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

Classification of Poincar´ e inequalities and PI-rectifiablity

Metric measure graph

Declare the edge to have length C ′(r(p) + r(q)) and associate measure proportional to µ(B(p, r(p))) + µ(B(q, r(q))). Doubling correspnds to doubling of X, and curves can be approximated by curves in the graph. Connectivity of X along the subset corresponds to connectivity

• f the graph. Need a connectivity condition involving sets.

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

Classification of Poincar´ e inequalities and PI-rectifiablity

Relationship to Muckenhoupt weights

Introduced by Muckenhoupt. “Quantitative absolute

• continuity. w ∈ A∞ if one of the following

1 For some 0 < δ < 1 there exists a 0 < ǫ < 1 such that for any

ball B ⊂ Rn and any E ⊂ Q w(E) ≤ ǫw(B) = ⇒ λ(E) ≤ δw(B)

2 There exists an 0 < α < 1 such that

λ(E) λ(B) ≤ C w(E) w(B) α

Classification of Poincar´ e inequalities and PI-rectifiablity

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 a

0 < ǫ < 1 such tha for any x, y ∈ X and every E ⊂ B(x, Cr) µ(E) ≤ ǫµ(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 there

exists a curve γ connecting the pair of points such that H1|γ(E) H1|γ(CB) ≤ C µ(E) µ(CB) α

Classification of Poincar´ e inequalities and PI-rectifiablity

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-Poincar´ e. (NHP)

Classification of Poincar´ e inequalities and PI-rectifiablity

Application

Theorem (E-B (2016)) If a D-doubling (X, d, µ) satisfies a (Φ, Ψ, C)-NHP, then it is a PI-space. I.e. It satisfies (1, q)-Poincar´ e inequality for some, possibly very large q.

Classification of Poincar´ e inequalities and PI-rectifiablity

Thank you!

Sylvester Eriksson-Bique ebs@cims.nyu.edu