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

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 Λ 1 / p p , where 1 < p < ∞ , and the Bloch space B , then the generalized Hilbert matrix H μ , 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 Λ 1 / p p , 1 < p < ∞ , by the space Λ 1 1 . 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 B 1 , 1 and the mixed norm space H ∞ , 1 , 1 . As immediate consequences, we obtain many results and some of them are new.