Texas 2009 nonlinear problems

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1 Texas 2009 nonlinear problems Workshop in Analysis and Probability August 2009, Texas A & M University, last update 07/28/2018 Abstract: This is a collection of problems suggested at two meetings of the seminar Nonlinear geometry of Banach spaces which took place during the 2009 Workshop in Analysis and Probability at Texas A & M University. Participants of the seminar: F. Baudier, J. Chavez-Dominguez, D. Dosev, T. Figiel, W. B. Johnson, P. W. Nowak, M. I. Ostrovskii, B. Randrianantoanina, B. Sari, G. Schechtman. Contents 1 First meeting (August 5, 2009) 1 2 Second meeting (August 10, 2009) Ostrovskii s talk and problems Johnson s modifications of problems from Section Johnson s problems on characterization of spaces with no cotype Arens-Eells spaces on expanders First meeting (August 5, 2009) Problem 1.1 (W. B. Johnson) Find a purely metric characterization of reflexivity or Radon-Nikodým property. Comment (02/13/2011). It is known ([CK09, Corollary 1.7], [Ost11, Theorem 3.6] that the Laakso [Laa00, Laa02] space does not admit a bilipschitz embeddings into a Banach space with the Radon-Nikodým property, but the converse does not seem to hold (see [Ost11]). Also, it is known that the infinite diamond does not admit a bilipschitz embedding into a Banach space with the Radon-Nikodým property, but the converse does not hold [Ost11, Corollary 3.3]. Comment (04/04/2014). Reflexivity was characterized in [Ost14+a]. The characterization is of the following type: there is a set of pairs S in l 1 l 1 such that a Banach space X is nonreflexive if and only if l 1 admits an embedding into X which satisfies the bilipschitz condition on pairs of S (but possibly not on other pairs). An if and only if metric characterization of the Radon-Nikodým property was found in [Ost14]. The characterization is terms of thick families of geodesics defined as follows. Let u and v be two elements in a metric space (M, d M ). A family T of uv-geodesics is 1

2 called thick if there is α > 0 such that for every g T and for every finite collection of points r 1,..., r n in the image of g, there is another uv-geodesic g satisfying the conditions: (1) The image of g also contains r 1,..., r n. (2) Possibly there are some more common points of g and g. (3) We can find a sequence 0 < s 1 < q 1 < s 2 < q 2 < < s m < q m < s m+1 < d M (u, v), such that g(q i ) = g(q i ) (i = 1,..., m) are common points containing r 1,..., r n, and the images g(s i ) and g(s i ) are distinct and the sum of deviations over them is nontrivially large in the sense that m+1 i=1 d M(g(s i ), g(s i )) α. The characterization of [Ost14] is: A Banach space X does not have the Radon- Nikodým property if and only if there exists a metric space M X containing a thick family T X of geodesics which admits a bilipschitz embedding into X. On the other hand it was shown in [Ost14+a] that for each thick family of geodesics there is a non-rnp Banach space which does not admit a uniformly bilipschitz embedding of it. In addition, in [Ost14+b] it was shown that a geodesic metric space which does not admit bilipschitz embeddings into Banach spaces with the Radon-Nikodým property does not necessarily contain a bilipschitz image of a thick family of geodesics. More precisely, the Heisenberg group with its subriemannian metric is an example of such metric space. Comment (07/28/2018). Motakis and Schlumprecht [MS17] found another metric characterization of reflexivity. The main result of their paper is based on the following definitions. The Schreier families S α are families of finite subsets of N, defined using transfinite induction for all α < ω 1 as follows: S 0 = { {n} : n N } { }. If α = γ + 1, that is, α is not a limit ordinal, then S α = { n j=1 E j : n min(e 1 ), E 1 < E 2 <... < E n, E j S γ, j = 1, 2,..., n}. Finally, if α is a limit ordinal we choose a fixed sequence ( λ(α, n) : n N ) [1, α) which increases to α and set S α = {E : k min(e), with E S λ(α,k) }. After that the authors define repeated averages on Schreier sets, which are positive elements of the unit sphere of l 1, indexed by pairs (α, A), where α is a countable ordinal and A is a maximal element of S α (the definition is too technical to reproduce it here). These averages are used to introduce two distances on S α. (1) The weighted tree distance on S α. For A, B in S α let C be the largest common initial segment of A and B, and then let d 1,α (A, B) = z (α,a) (a) + z (α,b) (b). a A\C b B\C (2) The weighted interlacing distance on S α. For A, B S α, say A = {a 1, a 2,..., a l } and B = {b 1, b 2,..., b m }, with a 1 < a 2 <... < a l and b 1 < b 2 <... < b m, we put a 0 = b 0 = 0 and a l+1 = b m+1 =, and define d,α (A, B) = max z (α,a) (a) + max z (α,b) (b). i=1,...,m+1 i=1,...,l+1 a A,b i 1 <a<b i b B,a i 1 <b<a i The main result of the paper: A Banach space X is reflexive if and only if there is α < ω 1, so that there is no mapping Φ : S α X for which cd,α (A, B) Φ(A) Φ(B) Cd 1,α (A, B) for all A, B S α. 2

3 2 Second meeting (August 10, 2009) 2.1 Ostrovskii s talk and problems The main motivation for the results of the talk is the problem: Let M be a metric space with bounded geometry which is not coarsely embeddable into a Hilbert space. Does it follow that M contains weakly a sequence of expanders (see [GK04], [Ost09a], [Tes09])? Comment (07/28/2018). This problem was solved in the negative by Arzhantseva and Tessera [AT15] using relative property (T) and the corresponding property of relative expansion. Talk was based on [Ost09b] where expansion properties of structures discovered in [Ost09a] and [Tes09] were studied. At the end of the talk the following problems were suggested: (1) Suppose that we have a sequence {G n } of graphs of uniformly bounded degrees and h > 0 (which does not depend on n) such that in each G n sets of diameters n are h-expanding. (The number of vertices in G n and its diameter can be significantly larger than n.) Can we find in it a substructures H n G n which are weak expanders? Remark. One of the classes of graphs satisfying the described condition are families of graphs of fixed degree 3 and indefinitely growing girth. (2) Let {G n } n=1 be a family of graphs satisfying the conditions of (1). Does it follow that {G n } are not uniformly coarsely embeddable into L 1? Comment (09/05/2011). Both problems were (implicitly) solved in the negative in [AGS12]. See also [Ost12] for more on the construction of [AGS12]. 2.2 Johnson s modifications of problems from Section 2.1 (a) What kind of expansion of metric spaces/graphs implies that they cannot be uniformly coarsely embedded into a Hilbert space? (b) Is there a sequence of finite metric spaces whose uniformly coarse embeddability into a metric space M is equivalent to the fact that M is not coarsely embeddable into a Hilbert space? 2.3 Johnson s problems on characterization of spaces with no cotype 1. Find a sequence of graphs with uniformly bounded degrees whose uniformly bilipschitz embeddability into a Banach space X is equivalent to the statement: X has no cotype. The same for type. Comment (02/13/2011). Problem 1 was solved in [Ost11]. Comment (09/05/2011). See [Ost13] for further results in the same direction. 2. Find a sequence of spaces with bounded geometry whose uniformly coarse embeddability into a Banach space X is equivalent to the statement: X has no cotype. 3

4 2.4 Arens-Eells spaces on expanders (M. I. Ostrovskii) Let {G n } be a family of expanders and {A n } be the sequence of the corresponding Arens-Eells spaces. Is it true that {A n } contain l k of growing dimensions? See [Kal08], [Ost13b, Chapter 10], and [Wea18] for basic facts on Arens-Eells spaces. References [AGS12] G. Arzhantseva, E. Guentner, J. Špakula, Coarse non-amenability and coarse embeddings, Geom. Funct. Anal., 22 (2012), 22 36; arxiv: [AT15] G. Arzhantseva, R. Tessera, Relative expanders. Geom. Funct. Anal. 25 (2015), no. 2, [Bau07] F. Baudier, Metrical characterization of super-reflexivity and linear type of Banach spaces, Archiv Math., 89 (2007), no. 5, [Bou86] J. Bourgain, The metrical interpretation of superreflexivity in Banach spaces, Israel J. Math., 56 (1986), no. 2, [CK09] [GK04] [JS09] J. Cheeger, B. Kleiner, Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property, Geom. Funct. Anal., 19 (2009), no. 4, E. Guentner, J. Kaminker, Geometric and analytic properties of groups, in: Noncommutative geometry, Ed. by S. Doplicher and R. Longo, Lecture Notes Math., 1831 (2004), W. B. Johnson, G. Schechtman, Diamond graphs and super-reflexivity, Journal of Topology and Analysis, 1 (2009), [Kal08] N. J. Kalton, The nonlinear geometry of Banach spaces, Rev. Mat. Complut., 21 (2008), no. 1, [Laa00] [Laa02] [MS17] [NR03] T. J. Laakso, Ahlfors Q-regular spaces with arbitrary Q > 1 admitting weak Poincare inequality, Geom. Funct. Anal., 10 (2000), no. 1, T. J. Laakso, Plane with A -weighted metric not bilipschitz embeddable into R N, Bull. London Math. Soc., 34 (2002), P. Motakis, Th. Schlumprecht, A metric interpretation of reflexivity for Banach spaces. Duke Math. J. 166 (2017), no. 16, I. Newman, Yu. Rabinovich, Lower bound on the distortion of embedding planar metrics into Euclidean space, Discrete Comput. Geom. 29 (2003), no. 1, [Ost09a] M. I. Ostrovskii, Coarse embeddability into Banach spaces, Topology Proceedings, 33 (2009), ; arxiv: [Ost09b] [Ost11] M. I. Ostrovskii, Expansion properties of metric spaces not admitting a coarse embedding into a Hilbert space, Comptes rendus de l Academie bulgare des Sciences, 62 (2009), , Expanded version: arxiv: M. I. Ostrovskii, On metric characterizations of some classes of Banach spaces, C. R. Acad. Bulgare Sci., 64 (2011), no. 6, The final version is available free of charge from the publisher ( [Ost12] M. I. Ostrovskii, Low-distortion embeddings of graphs with large girth, J. Funct. Anal., 262 (2012), ; arxiv: [Ost13] M. I. Ostrovskii, Different forms of metric characterizations of classes of Banach spaces, Houston J. Math., 39 (2013), no. 3, [Ost13b] M. I. Ostrovskii, Metric Embeddings: Bilipschitz and Coarse Embeddings into Banach Spaces, de Gruyter Studies in Mathematics, 49. Walter de Gruyter & Co., Berlin, [Ost14] M. I. Ostrovskii, Radon-Nikodým property and thick families of geodesics, J. Math Anal. Appl., 409 (2014), no. 2, [Ost14+a] M. I. Ostrovskii, On metric characterizations of the Radon-Nikodým and related properties of Banach spaces, Journal of Topology and Analysis, 6 (2014),

5 [Ost14+b] M. I. Ostrovskii, Metric spaces nonembeddable into Banach spaces with the Radon-Nikodým property and thick families of geodesics, Fundamenta Mathematicae, 227 (2014), [Tes09] [Wea18] R. Tessera, Coarse embeddings into a Hilbert space, Haagerup property and Poincaré inequalities, Journal of Topology and Analysis, 1 (2009), N. Weaver, Lipschitz algebras, Second edition, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ,