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