Classification of Poincar e inequalities and PI-rectifiablity - PowerPoint PPT Presentation

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. | f ( x ) − f ( y ) | Lip f ( x ) = lim sup d ( x , y ) x � = y → x

Classification of Poincar´ e inequalities and PI-rectifiablity Poincar´ e inequality For every Lipschitz f : X → R � 1 � p Lip f p d µ | f − f B ( x , r ) | d µ ≤ Cr . (1) B ( x , r ) B ( x , C ′ r ) 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¨ orn, 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 R n . 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 , ˆ ρ p d µ, inf ρ B 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 only weaker conditions can be obtained directly.

Classification of Poincar´ e inequalities and PI-rectifiablity Obligatory Slide Theorem (Rademacher’s theorem) Every Lipschitz f : R n → 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 U i , Lip-functions φ i : X → R n i µ ( X \ � U i ) = 0 Every Lip function f : X → R N , for every i and almost every x ∈ U i has a unique derivative df i ( x ): R n i → R N s.t. f ( y ) − f ( x ) = df i ( x )( φ i ( y ) − φ i ( 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 U i , Lip-functions φ i : X → R n i µ ( X \ � U i ) = 0 V is an arbitrary RNP-Banach space ( L p , l p , c 0 , NOT L 1 ) Every Lip function f : X → V , for every i and almost every x ∈ U i has a unique derivative df i ( x ): R n i → V s.t. f ( y ) − f ( x ) = df i ( x )( φ i ( y ) − φ i ( 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 U i ? (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 U i to “connected” metric spaces U i by glueing a “tree-like” graph to it, which approximates a neighborhood in X . How to establish Poincar´ e inequalities for U i 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 | ≤ δ r

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 . e holds for p > 1 Once α is identified, 1 / p -Poincar´ 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 , r 0 ) -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 U i ⊂ X , which are “relatively” doubling and “relatively” (and locally) ( C , δ, ǫ )-connected. Need a way to find something to glue to U i to get ( C ′ , δ ′ , ǫ ′ )-connectivity of a larger space U i , 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 r 0 > 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.