Generalized Hilbert operator acting on Bergman spaces

TL;DR

This paper characterizes measures for which a generalized Hilbert operator acts boundedly or compactly between various analytic function spaces, extending previous results and identifying Hilbert-Schmidt class conditions.

Contribution

It generalizes the study of the integral Hilbert operator to a broader class of measures and spaces, providing new boundedness, compactness, and Hilbert-Schmidt criteria.

Findings

Characterization of measures for boundedness and compactness of the operator

Extension of results to Bergman spaces with 1 ≤ p ≤ 2

Determination of the Hilbert-Schmidt class on A^2 for all α > -1

Abstract

Let $\mu$ be a positive Borel measure on $[0,1)$. If $f \in H(\mathbb{D})$ and $\alpha>-1$, the generalized integral type Hilbert operator defined as follows: $$\mathcal{I}_{\mu_{\alpha+1}}(f)(z)=\int^1_{0} \frac{f(t)}{(1-tz)^{\alpha+1}}d\mu(t), \ \ \ z\in \mathbb{D} .$$ The operator $\mathcal{I}_{\mu_{1}}$ has been extensively studied recently. In this paper, we characterize the measures $\mu$ for which $\mathcal{I}_{\mu_{\alpha+1}}$ is a bounded (resp., compact) operator acting between the Bloch space $\mathcal {B}$ and Bergman space $ A^{p}$, or from $A^{p}(0<p<\infty)$ into $ A^{q}(q\geq 1)$. We also study the analogous problem in Bergman spaces $A^{p}(1 \leq p\leq 2)$. Finally, we determine the Hilbert-Schmidt class on $A^{2}$ for all $\alpha>-1$.