Hilbert matrix on spaces of Bergman-type

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It is well known (see [8,14]) that the Libera operator L is bounded on the Besov space Hνp,q,α if and only if 0<κp,α,ν:=ν−α−1p+1. We prove unexpected results: the Hilbert matrix operator H, as well as the modified Hilbert operator H˜, is bounded on Hνp,q,α if and only if 0<κp,α,ν<1. In particular, H, as well as H˜, is bounded on the Bergman space Ap,α if and only if 1<α+2<p and is bounded on the Dirichlet space Dαp=A1p,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].