This is an announcement for the paper "Markov type and threshold embeddings" by Jian Ding, James R. Lee, and Yuval Peres.
Abstract: For two metric spaces $X$ and $Y$, say that $X$ {\em threshold-embeds into $Y$} if there exist a number $K > 0$ and a family of Lipschitz maps ${\varphi_{\tau} : X \to Y : \tau > 0 }$ such that for every $x,y \in X$, $$ d_X(x,y) \geq \tau \implies d_Y(\varphi_{\tau}(x),\varphi_{\tau}(y)) \geq |\varphi_{\tau}|_{\Lip} \tau/K,, $$ where $|\varphi_{\tau}|_{\Lip}$ denotes the Lipschitz constant of $\varphi_{\tau}$. We show that if a metric space $X$ threshold-embeds 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 $p$. The preceding result, together with Kwapien's theorem, is used to show that if a Banach space threshold-embeds into a Hilbert space then it is linearly isomorphic to a Hilbert space. This suggests some non-linear analogs of Kwapien's theorem. For instance, a subset $X \subseteq L_1$ threshold-embeds into Hilbert space if and only if $X$ has Markov type 2.
Archive classification: math.MG math.FA math.PR
Submitted from: jrl@cs.washington.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.6088
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