Generalized Hilbert Matrices Acting on Spaces that are Close to the Hardy Space H-1 and to the Space BMOA

摘要

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 p 1/p, where 1 < p < 8, and the Bloch space B, then the generalized Hilbert matrix H mu, induced by a positive Borel measure mu on the interval [ 0, 1), is a bounded operator from the space X into the space Y if and only if mu is a 1-logarithmic 1-Carleson measure. We improve this result by proving that the same conclusion holds if we replace the space p 1/p, 1 < p < 8, by the space 11. 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 8, 1,1. As immediate consequences, we obtain many results and some of them are new.