Markov type constants of Wasserstein spaces

For $p > 1$, $T \in \mathbb{N}$ and a metric space $X$ we debote by $M_p(X,T)$ the Markov type $p$ constant at time $T$ of $X$. The Markov type $p$ constant of $X$, denoted by $M_p(X)$ is defined by

$$M_p(X) = \sup_{T \in \mathbb{N}}M(X,T) \in [1,\infty].$$

We say that $X$ has Markov type $p$ if $M_p(X) < \infty$. The notion of Markov type was introduced by K. Ball  [1] and has two main applications:

(1) Partial Lipshitz maps from a metric space $X$ having Markov type $p$ are extendable to Lipshitz maps of the whole space.}
(2) Obstruction to bi-Lipshitz embeddability. Metric space $X$ can not be embedded into metric space $Y$ with a bi-Lipshitz distortion less then $\sqrt{\frac{M_p(X,T)}{M_p(Y,T)}}$.}

In the talk I'm going to present an idea which allows to compute Markov type constants for Wasserstein spaces. Let $\mathcal{P}_p(\mathbb{R}^d)$ denotes the Wasserstein space over the Euclidean space $\mathbb{R}^d$. We obtain following estimates.

Theorem ([3], Corollary 3) For every $p \in (2,\infty)$ and $T,d \in \mathbb{N}$ we have

(1) {$M_p(\mathcal{P}_p(\mathbb{R}^d), T) \le 16 d^{1/2 - 1/p}p^{1/2}T^{1/2-1/p},$\label{l1}}
(2) {$M_2(\mathcal{P}_p(\mathbb{R}^d)) \le 4d^{1/2 - 1/p}\sqrt{p-1}$.\label{l2}}

As observed by A. Andoni, A. Naor and O. Neiman the upper bound for $M_p(mathcal{P}_p(\mathbb{R}^d), T)$  implies certain restriction on embeddability of snowflakes into $\mathcal{P}_p(\mathbb{R}^d)$. Theorem(\ref{l2}) provides an extension theorem for partial Lipshitz maps from $\mathcal{P}_p(\mathbb{R}^d)$ into CAT(0) spaces, uniformly convex Banach spaces or more generally metric spaces with metric Markov cotype $2$(See  [2][Theorem 1.11, Corollary 1.13]{MN}).
 

References
[1] K. Ball. Markov chains, Riesz transforms and Lipschitz maps. Geometric and Functional Analysis GAFA, 2(2):137–172, 1992.
[2] Manor Mendel and Assaf Naor. Spectral calculus and lipschitz extension for barycentric metric spaces. Analysis and Geometry in Metric Spaces, 1:163–199.
[3] Vladimir Zolotov. Markov type constants, flat tori and wasserstein spaces. arXivpreprint arXiv:1610.04886, 2016.
 

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