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 $f(0)=0$, $|f^{\prime }(t)|\lesssim |t{|}^{\alpha }$, and let $\lambda \in {\ell }^r$ be real with $\Vert \lambda {\Vert }_{{\ell }^{\infty }}\le 1$. We determine the pairs $0<p,q\le \infty$ 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 \[ \frac1p\leq\fracα{r} +\min\!\left\{1,\frac1q\right\}. \] 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.