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We establish a rather unexpected and simple criterion for the boundedness ofSchur multipliers $S_M$ on Schatten $p$-classes which solves a conjectureproposed by Mikael de la Salle. Given $1 < p < \infty$, a simple form our mainresult 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-Mikhlinmultiplier 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 leadsto new Littlewood-Paley characterizations of $S_p$-norms and strongapplications in harmonic analysis for nilpotent and high rank simple Lie groupalgebras.

Schur multipliers in Schatten-von Neumann classes