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 $Λ^{p}_{\frac{1}{p}}$ 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 $Λ^{1.*}_{1}$), which is strictly smaller than $\cap_{p>1}Λ^{p}_{\frac{1}{p}}$. We also provide explicit upper and lower bounds for the norm of the Hilbert matrix $\mathcal {H}$ acting from $H^{\infty}$ to $Λ^{1.*}_{1}$. Additionally, we also characterize the positive Borel measures $μ$ such that the generalized Hilbert matrix operator $\mathcal {H}_μ$ 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 $\mathcal {H}_μ$ on Hardy spaces.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• Lemma 2.2
• Remark 2.3
• Remark 2.4
• Lemma 3.1