Hilbert matrix operator on bound analytic functions

Abstract

It is well known that the Hilbert matrix operator $\mathcal{H}$ is bounded from $H^{\infty }$ to the mean Lipschitz spaces $\hat{I}{›}_{\frac{1}{p}}^p$ for all $1<p<\infty$. In this paper, we prove that the range of Hilbert matrix operator $\mathcal{H}$ acting on $H^{\infty }$ is contained in certain Zygmund-type space (denoted by $\hat{I}{›}_1^{1.\ast }$), which is strictly smaller than ${\cap }_{p>1}\hat{I}{›}_{\frac{1}{p}}^p$. We also provide explicit upper and lower bounds for the norm of the Hilbert matrix $\mathcal{H}$ acting from $H^{\infty }$ to $\hat{I}{›}_1^{1.\ast }$. Additionally, we also characterize the positive Borel measures $\hat{I}¼$ such that the generalized Hilbert matrix operator ${\mathcal{H}}_{\hat{I}}¼$ is bounded from $H^{\infty }$ to the Hardy space $H^q$. This part is a continuation of the work of Chatzifountas, Girela and Peláez [J. Math. Anal. Appl. 413 (2014) 154--168] regarding ${\mathcal{H}}_{\hat{I}}¼$ on Hardy spaces.