Arazy's conjecture concerning Schur multipliers: revisited and resolved

Let ${\mathcal{S}}^r$ denote the Schatten–von Neumann class and let $S_{{\mathrm{\Psi }}_{f,\lambda }}$ be the Schur–Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $\lambda$. Let $0<\alpha ,r<\mathrm{\infty }$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $\mathrm{\mid }{f}^{\mathrm{\prime }}(t)\mathrm{\mid }\lesssim \mathrm{\mid }t{\mathrm{\mid }}^{\alpha }$, and let $\lambda \in {\mathrm{\ell }}^r$ be real with $\mathrm{\parallel }\lambda {\mathrm{\parallel }}_{{\mathrm{\ell }}^{\mathrm{\infty }}}\le 1$. We determine the pairs $0<p,q\le \mathrm{\infty }$ for which $S_{{\mathrm{\Psi }}_{f,\lambda }}:{\mathcal{S}}^q\to {\mathcal{S}}^p$ is bounded for every such $f$ and $\lambda$. Our principal new positive estimates treat $q=\mathrm{\infty },1$ and $0<p<1$. Precisely, we show that the boundedness holds exactly when $$\frac{1}{p}\le \frac{\alpha }{r}+\min \text{\hspace{0.17em}⁣}\{1,\frac{1}{q}\}\mathrm{.}$$ 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.