Norm estimates for the Hilbert matrix operator on weighted Bergman spaces

We study the Hilbert matrix operator H and a related integral operator T acting on the standard weighted Bergman spaces A α p . We obtain an upper bound for T , which yields the smallest currently known explicit upper bound for the norm of H for − 1 < α < 0 and 2 + α < p < 2 ( 2 + α ) . We also calculate the essential norm for all p > 2 + α > 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.