This is an announcement for the paper "Metric ${X}_p$ inequalities" by Assaf Naor and Gideon Schechtman.

Abstract: We show that if $m,n\in \mathbb{N}$ and $k\in {1,\ldots, n}$ satisfy $m\ge \frac{n^{3/2}}{\sqrt{k}}$ then for every $p\in [2,\infty)$ and $f:\mathbb{Z}_{4m}^n\to \mathbb{R}$ we have \begin{equation} \frac{1}{\binom{n}{k}}\sum_{\substack{S\subseteq {1,\ldots,n}\|S|= k}}\frac{\mathbb{E}\left[\big|f\big(x+2m\sum_{j\in S} \varepsilon_j e_j\big)-f(x)\big|^p\right]}{m^p}\lesssim_p \frac{k}{n}\sum_{j=1}^n\mathbb{E}\big[\left| f(x+e_j)-f(x)\right|^p\big]+\left(\frac{k}{n}\right)^{\frac{p}{2}} \mathbb{E}\big[\left|f\left(x+ \varepsilon{e}\right)-f(x)\right|^p\big], \end{equation} where the expectation is with respect to $(x,\varepsilon)\in \mathbb{Z}_{4m}^n\times {-1,1}^n$ chosen uniformly at random and $e_1,\ldots e_n$ is the standard basis of $\mathbb{Z}_{4m}^n$. The above inequality is a nonlinear extension of a linear inequality for Rademacher sums that was proved by Johnson, Maurey, Schechtman and Tzafriri in 1979. We show that for the above statement to hold true it is necessary that $m$ tends to infinity with $n$. The formulation (and proof) of the above inequality completes the long-standing search for bi-Lipschitz invariants that serve as an obstruction to the nonembeddability of $L_p$ spaces into each other, the previously understood cases of which were metric notions of type and cotype, which fail to certify the nonembeddability of $L_q$ into $L_p$ when $2<q<p$. Among the consequences of the above inequality are new quantitative restrictions on the bi-Lipschitz embeddability into $L_p$ of snowflakes of $L_q$ and integer grids in $\ell_q^n$, for $2<q<p$. As a byproduct of our investigations, we also obtain results on the geometry of the Schatten $p$ trace class $S_p$ that are new even in the linear setting.

Archive classification: math.FA math.MG math.OA

Submitted from: naor@cims.nyu.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1408.5819

or

http://arXiv.org/abs/1408.5819