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 of our main result for $\mathbf{R}^n \times \mathbf{R}^n$ matrices reads 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 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.
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