Generalized Hilbert operator acting on Bergman spaces

Abstract

Let $\hat{I}¼$ be a positive Borel measure on $[0,1)$. If $f\in H(\mathbb{D})$ and $\hat{I}\pm >-1$, the generalized integral type Hilbert operator defined as follows: $${\mathcal{I}}_{\hat{I}{¼}_{\hat{I}\pm +1}}(f)(z)={\int }_0^1\frac{f(t)}{(1-tz{)}^{\hat{I}\pm +1}}d\hat{I}¼(t),z\in \mathbb{D}.$$ The operator ${\mathcal{I}}_{\hat{I}{¼}_1}$ has been extensively studied recently. In this paper, we characterize the measures $\hat{I}¼$ for which ${\mathcal{I}}_{\hat{I}{¼}_{\hat{I}\pm +1}}$ 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 $A^q(q\ge 1)$. We also study the analogous problem in Bergman spaces $A^p(1\le p\le 2)$. Finally, we determine the Hilbert-Schmidt class on $A^2$ for all $\hat{I}\pm >-1$.