Markov type and threshold embeddings

作者：Ding Jian^*; Lee James R; Peres Yuval

来源：Geometric and Functional Analysis, 2013, 23(4): 1207-1229.

DOI：10.1007/s00039-013-0234-7

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 {psi(tau) : X -> Y : tau > 0} such that for every x, y epsilon X ,
d(X)(x, y) >= tau double right arrow d(Y) (psi(tau)(x), psi(tau) (y)) >= parallel to psi(tau)parallel to Lip(tau)/K,
where parallel to psi(tau) parallel to Lip denotes the Lipschitz constant of psi(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 subset of L-1 threshold-embeds into Hilbert space if and only if X has Markov type 2.

• 出版日期2013-8
• 单位Microsoft