Hilbert矩阵算子到Bergman空间

中文摘要 AI 辅助

我们证明了当$\alpha\geq 2$时，Hilbert矩阵算子$H:A^2_{\ln^{\alpha}} \to A^2$是有界的，并且当$\alpha<2$时，$H$不将$A^2_{\ln^{\alpha}}$映射到$A^2$，从而解决了B. Karapetrović和M. Jevtić提出的一个开放问题。当$\alpha=2$时的有界性作为更一般命题的推论：Hilbert矩阵$H:\ell_v^p \to \ell_w^p$是有界的，其中$v(n) = (\ln n)^p / n$且$w(n) = 1/n$。

英文摘要

We prove that the Hilbert matrix operator $H:A^2_{\ln^α} \to A^2$ is bounded when $α\geq 2$ and that $H$ does not map $A^2_{\ln^α} $ into $A^2$ when $α<2$, solving an open problem posed by B. Karapetrović and M. Jevtić. The boundedness when $α=2$ is shown as a corollary to the more general statement: the Hilbert matrix $H:\ell_v^p \to \ell_w^p$ is bounded, where $v(n) = (\ln n)^p / n$ and $w(n) = 1/n$.

发表机构

• Department of Mathematical Sciences, Faculty of Information Technology and Electrical Engineering, NTNU – Norwegian University of Science and Technology（挪威科技大学信息与电气工程学院数学科学系）

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