Markov type and threshold embeddings

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 ${{varphi_{tau} : X to Y : tau > 0}}$ such that for every ${x,y in X}$ , $$d_X(x, y) geq tau implies d_Y(varphi_tau (x),varphi_tau (y)) geq |{varphi}_tau|_{rm Lip}tau/K,$$ where ${|{varphi}_{tau}|_{rm Lip}}$ 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. Our results suggest 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.