Markov type and threshold embeddings

Abstract

For two metric spaces X and Y , say that X threshold-embeds into Y if there exist a number K > 0 and a family of Lipschitz maps {ϕ τ : X → Y : τ > 0} such that for every x, y ∈ X,where ϕ τ Lip denotes the Lipschitz constant of ϕ τ . We show that if a metric space X thresholdembeds into a Hilbert space, then X has Markov type 2. As a consequence, planar graph metrics and doubling metrics have Markov type 2, answering questions of Naor, Peres, Schramm, and Sheffield. More generally, if a metric space X threshold-embeds into a p-uniformly smooth Banach space, then X has Markov type p. Our results suggest some non-linear analogs of Kwapien's theorem. For instance, a subset X ⊆ L 1 threshold-embeds into Hilbert space if and only if X has Markov type 2.

Metric Inequalities

“…The search for metric invariants that explain the first range in (1) was an important impetus in the development of the theory of type of metric spaces, with notable contributions by Enflo [29,30,31], Bourgain-Milman-Wolfson [19], Pisier [78] and Ball [9]; see also [71,70,60,68,73,35,69,27,39,67]. The search for metric invariants that explain the second range in (1) was an important impetus in the development of the theory of cotype of metric spaces; see the work of Mendel and Naor [61] as well as [9,34,63,64].…”

Diffusive estimates for random walks on stationary random graphs of polynomial growth

“…Let us focus now on condition (1) since it is the difficult one to verify. In order to control the speed of the random walk started at a uniformly random point of B G ρ (r n ), we construct a family of mappings {F k } from B G ρ (r n ) into a Hilbert space and use the martingale methods of [NPSS06,DLP13] to derive bounds on the speed. The following statement is a slightly weaker version of Lemma 2.3 in Section 2.1.…”

“…It is proved in [DLP13] that if such a threshold embedding exists, then M 2 (X, d) O(D) (recall the definition of Markov type from Section 1.3). Bounding the graphic Markov type is substantially easier.…”

Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

“…It was also proved in [50] that trees, hyperbolic groups, complete simply connected Riemannian manifolds of pinched sectional curvature and Laakso graphs all have Markov type 2, and that spaces that admit a padded random partition (see [36]), in particular doubling metric spaces and planar graphs, have Markov type for all ∈ (0 2). In [10] it was shown that series parallel graphs have Markov type 2, and finally in the recent work [13] it was shown that spaces that admit a padded random partition have Markov type 2. Thus, in particular, doubling spaces and planar graphs have Markov type 2.…”

“…Thus, in particular, doubling spaces and planar graphs have Markov type 2. In [52] it was shown that spaces with finite Nagata dimension admit a padded random partition, and so by [13] ) is a doubling space, planar graph, or a space with finite Nagata dimension, then (X Y ) is finite. These results were previously proved in [36] via the method of random partitions (Lipschitz extension for spaces of bounded Nagata dimension was previously treated in [34] and only later it was shown in [52] that they admit a padded random partition and therefore the corresponding extension results are a special case of [36]).…”