Hilbert matrix on spaces of Bergman-type

Abstract

It is well known (see [8,14]) that the Libera operator L is bounded on the Besov space H<inf>ν</inf>^p,q,α if and only if 0<κ<inf>p,α,ν</inf>:=ν−α−1/p+1. We prove unexpected results: the Hilbert matrix operator H, as well as the modified Hilbert operator H˜, is bounded on H<inf>ν</inf>^p,q,α if and only if 0<κ<inf>p,α,ν</inf><1. In particular, H, as well as H˜, is bounded on the Bergman space A^p,α if and only if 1<α+2<p and is bounded on the Dirichlet space D<inf>α</inf>^p=A<inf>1</inf>^p,α if and only if max⁡{−1,p−2}<α<2p−2. Our results are substantial improvement of [11, Theorem 3.1] and of [6, Theorem 5].