Hilbert Matrix and Its Norm on Weighted Bergman Spaces

Abstract

It is well known that the Hilbert matrix ${\mathrm {H}}$ is bounded on weighted Bergman spaces $A^p_\alpha$ if and only if $1<\alpha +2<p$ with the conjectured norm $\pi /\sin \frac{(\alpha +2)\pi }{p}$. The conjecture was confirmed in the case when $\alpha =0$ and also in the case when $\alpha >0$ and $p\ge 2(\alpha +2)$, which reduces the conjecture in the case when $\alpha >0$ to the interval $\alpha +2<p<2(\alpha +2)$. In the remaining case when $-1<\alpha <0$ and $p>\alpha +2$ there has been no progress so far in proving the conjecture, moreover, there is no even an explicit upper bound for the norm of the Hilbert matrix ${\mathrm {H}}$ on weighted Bergman spaces $A^p_\alpha$. In this paper we obtain results which are better than known related to the validity of the mentioned conjecture in the case when $\alpha >0$ and $\alpha +2<p<2(\alpha +2)$. On the other hand, we also provide for the first time an explicit upper bound for the norm of the Hilbert matrix ${\mathrm {H}}$ on weighted Bergman spaces $A^p_\alpha$ in the case when $-1<\alpha <0$ and $p>\alpha +2$.