The Hilbert matrix operator into the Bergman space

Abstract

We prove that the Hilbert matrix operator $H:A_{{\ln }^{\alpha }}^2\to A^2$ is bounded when $\alpha \ge 2$ and that $H$ does not map $A_{{\ln }^{\alpha }}^2$ into $A^2$ when $\alpha <2$, solving an open problem posed by B. Karapetrović and M. Jevtić. The boundedness when $\alpha =2$ 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 $w(n)=1/n$.