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

Abstract

It is known that if X and Y are spaces of holomorphic functions in the unit disc $\mathbb {D}$, which are between the mean Lipschitz space $\Lambda ^p_{1/p}$, where $1<p<\infty$, and the Bloch space $\mathcal {B}$, then the generalized Hilbert matrix $\mathcal {H}_\mu$, 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 $\Lambda ^p_{1/p}$, $1<p<\infty$, by the space $\Lambda ^1_1$. Also we prove that the same conclusion holds if X and Y are spaces of holomorphic functions in $\mathbb {D}$, which are between the Besov space $\mathcal {B}^{1,1}$ and the mixed norm space $H^{\infty ,1,1}$. As immediate consequences, we obtain many results and some of them are new.