Schur multipliers in Schatten-von Neumann classes

Abstract

We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers $S_M$ on Schatten $p$-classes which solves a conjecture proposed by Mikael de la Salle. Given $1<p<\infty$, a simple form our main result reads for ${\mathbf{R}}^n\times {\mathbf{R}}^n$ matrices as follows $$\Vert S_M:S_p\to S_p{\Vert }_{\mathrm{cb}}\lesssim \frac{p^2}{p-1}\sum _{|\gamma |\le [\frac{n}{2}]+1}\Vert |x-y{|}^{|\gamma |}\{|{\partial }_x^{\gamma }M(x,y)|+|{\partial }_y^{\gamma }M(x,y)|\}{\Vert }_{\infty }.$$ In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders $\sigma >\frac{n}{2}$ as well. It trivially includes Arazy's conjecture for $S_p$-multipliers and extends it to $\alpha$-divided differences. It also leads to new Littlewood-Paley characterizations of $S_p$-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.

To appear in Annals of Mathematics. Affiliations corrected