Hilbert matrix operator on bound analytic functions

TL;DR

This paper investigates the boundedness and range of the Hilbert matrix operator on bounded analytic functions, identifying a new target space and providing bounds for the operator's norm, along with characterizations related to measures and Hardy spaces.

Contribution

It establishes that the Hilbert matrix operator maps bounded analytic functions into a specific Zygmund-type space and characterizes measures for boundedness into Hardy spaces, extending previous results.

Findings

Range of $ ext{H}$ is contained in a smaller Zygmund-type space.

Explicit bounds for the operator norm are provided.

Characterization of measures for boundedness into Hardy spaces.

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.