Arazy's conjecture concerning Schur multipliers: revisited and resolved

Abstract

Let $\mathcal{S}^ r$ denote the Schatten--von Neumann class and let $S_{\Psi_{f,\lambda}}$ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^\alpha$, and let $\lambda\in\ell^r$ be real with $\|\lambda\|_{\ell^\infty}\leq1$. We determine the pairs $0<p,q\leq\infty$ for which $S_{\Psi_{f,\lambda}}:\mathcal{S}^ q\to\mathcal{S}^ p$ is bounded for every such $f$ and $\lambda$. Our principal new positive estimates treat $q=\infty,1$ and $0<p<1$. Precisely, we show that the boundedness holds exactly when \[ \frac1p\leq\frac{\alpha}{r} +\min\!\left\{1,\frac1q\right\}. \] In particular, this resolves the untreated cases in [Arazy, PAMS, 1982] and [Potapov, Sukochev, Tomskova, Adv. Math., 2015]. Our method also delivers a new proof of the main results in just cited papers.