Discrete Riesz transforms and sharp metric $X_p$ inequalities

Abstract: $ \renewcommand{\subset}{\subseteq} \newcommand{\N}{\mathbb N} $For $p\in [2,\infty)$ the metric $X_p$ inequality with sharp scaling parameter is proven here to hold true in $L_p$. The geometric consequences of this result include the following sharp statements about embeddings of $L_q$ into $L_p$ when $2< q<p<\infty$: the maximal $\theta\in (0,1]$ for which $L_q$ admits a bi-$\theta$-H\"older embedding into $L_p$ equals $q/p$, and for $m,n\in \N$ the smallest possible bi-Lipschitz distortion of any embedding into $L_p$ of the grid ${1,\ldots,m}^n\subset \ell_q^n$ is bounded above and below by constant multiples (depending only on $p,q$) of the quantity $\min{n^{(p-q)(q-2)/(q^2(p-2))}, m^{(q-2)/q}}$.