Schur multipliers in Schatten-von Neumann classes

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 $$\big\| S_M: S_p \to S_p \big\|_{\mathrm{cb}} \lesssim \frac{p^2}{p-1} \sum_{|\gamma| \le [\frac{n}{2}] +1} \Big\| |x-y|^{|\gamma|} \Big\{ \big| \partial_x^\gamma M(x,y) \big| + \big| \partial_y^\gamma M(x,y) \big| \Big\} \Big\|_\infty.$$ In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the H\"ormander-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.