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 $\Lambda^{p}_{\frac{1}{p}}$ 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 $\Lambda^{1.*}_{1}$), which is strictly smaller than $\cap_{p\>1}\Lambda^{p}_{\frac{1}{p}}$. We also provide explicit upper and lower bounds for the norm of the Hilbert matrix $\mathcal {H}$ acting from $H^{\infty}$ to $\Lambda^{1.*}_{1}$. Additionally, we also characterize the positive Borel measures $\mu$ such that the generalized Hilbert matrix operator $\mathcal {H}{\mu}$ is bounded from $H^{\infty}$ to the Hardy space $H^{q}$. This part is a continuation of the work of Chatzifountas, Girela and Pel\'{a}ez [J. Math. Anal. Appl. 413 (2014) 154--168] regarding $\mathcal {H}{\mu}$ on Hardy spaces.