Discrete Riesz transforms and sharp metric $X_p$ inequalities

\renewcommand{\subset}{\subseteq} \newcommand{\N}{\mathbb N}For $p\in [2,\infty)$themetric$X_p$ 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 $2< q<p<\infty$:themaximal$\theta\in (0,1]$forwhich$L_q$ admits a bi-$\theta$-H"older embedding into $L_p$ equals $q\mathrm{/}p$, and for $m,n\in \mathbb{N}$ the smallest possible bi-Lipschitz distortion of any embedding into $L_p$ of the grid $\{1,\dots ,m{\}}^n\subset {\mathrm{\ell }}_q^n$ is bounded above and below by constant multiples (depending only on $p,q$) of the quantity $\min \{n^{(p-q)(q-2)\mathrm{/}(q^2(p-2))},m^{(q-2)\mathrm{/}q}\}$.