Arazy关于Schur乘子的猜想：重访与解决

中文摘要 AI 辅助

设$\mathcal{S}^r$表示Schatten-von Neumann类，$S_{\Psi_{f,\lambda}}$为Schur-Hadamard乘子，其符号是$f$沿$\lambda$的差商矩阵。设$0<\alpha,r<\infty$，$f\in C^1([-1,1])$满足$f(0)=0$，$|f'(t)|\lesssim |t|^\alpha$，且$\lambda\in\ell^r$为实数，满足$\\|\lambda\\|_{\ell^\infty}\leq1$。我们确定了所有满足$0<p,q\leq\infty$的配对，使得对于每个这样的$f$和$\lambda$，$S_{\Psi_{f,\lambda}}:\mathcal{S}^q\to\mathcal{S}^p$有界。我们主要的新正估计处理$q=\infty,1$和$0<p<1$的情况。精确地说，我们证明有界性恰好当\\[ \frac1p\leq\frac{\alpha}{r}+\min\\!\left\{1,\frac1q\right\} \\]时成立。特别地，这解决了[Arazy, PAMS, 1982]和[Potapov, Sukochev, Tomskova, Adv. Math., 2015]中未处理的情况。我们的方法也为上述引文中的主要结果提供了新的证明。

英文摘要

Let $\mathcal{S}^ r$ denote the Schatten--von Neumann class and let $S_{Ψ_{f,λ}}$ be the Schur--Hadamard multiplier whose symbol is the divided-difference matrix of $f$ along $λ$. Let $0<α,r<\infty$, let $f\in C^1([-1,1])$ satisfy $f(0)=0$, $|f'(t)|\lesssim |t|^α$, and let $λ\in\ell^r$ be real with $\|λ\|_{\ell^\infty}\leq1$. We determine the pairs $0<p,q\leq\infty$ for which $S_{Ψ_{f,λ}}:\mathcal{S}^ q\to\mathcal{S}^ p$ is bounded for every such $f$ and $λ$. 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.

发表机构

• Institute for Advanced Study in Mathematics of HIT, Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）
• School of Mathematics and Statistics, University of NSW（新南威尔士大学数学与统计学院）

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