Discrete Riesz transforms and sharp metric $X_p$ inequalities

Abstract

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 $2\lt q\lt p\lt \infty$: the maximal $\theta\in (0,1]$ for which $L_q$ admits a bi-$\theta$-Hölder embedding into $L_p$ equals $q/p$, and for $m,n\in \mathbb{N}$, 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 $\min\{n^{(p-q)(q-2)/(q^2(p-2))}, m^{(q-2)/q}\}$.