Generalized Hilbert Matrices Acting on Spaces that are Close to the Hardy Space H1 and to the Space BMOA

Abstract

It is known that if X and Y are spaces of holomorphic functions in the unit disc D, which are between the mean Lipschitz space Λ1/pp, where 1 < p< ∞, and the Bloch space B, then the generalized Hilbert matrix H<inf>μ</inf>, induced by a positive Borel measure μ on the interval [0, 1), is a bounded operator from the space X into the space Y if and only if μ is a 1-logarithmic 1-Carleson measure. We improve this result by proving that the same conclusion holds if we replace the space Λ1/pp, 1 < p< ∞, by the space Λ11. Also we prove that the same conclusion holds if X and Y are spaces of holomorphic functions in D, which are between the Besov space B1 , 1 and the mixed norm space H∞ , 1 , 1. As immediate consequences, we obtain many results and some of them are new.