Abstract of a paper by Jian Ding, James R. Lee, and Yuval Peres
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 or http://arXiv.org/abs/1208.6088
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alspach@math.okstate.edu