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...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2201.05511 |