Norm estimates for the Hilbert matrix operator on weighted Bergman spaces

Abstract

We study the Hilbert matrix operator H and a related integral operator T acting on the standard weighted Bergman spaces $A_{\alpha }^p$. We obtain an upper bound for T, which yields the smallest currently known explicit upper bound for the norm of H for $-1<\alpha <0$ and $2+\alpha <p<2(2+\alpha )$. We also calculate the essential norm for all $p>2+\alpha >1$, extending a part of the main result in [Adv. Math. 408 (2022) 108598] to the standard unbounded weights. It is worth mentioning that except for an application of Minkowski's inequality, the norm estimates obtained for T are sharp.