Hankel-Type Operator Acting on Hardy Spaces and Weighted Bergman Spaces

Abstract

Inspired by Xiao’s work about the Hankel measures for the weighted Bergman spaces, in this paper, if $\beta >0$ and the measure $\mu$ is a complex Borel measure on the unit disk ${\mathbb {D}}$, we define the Hankel type operator $K_{\mu ,\beta }$ by

$$\begin{aligned} K_{\mu ,\beta }:~f\longmapsto \int _{{\mathbb {D}}}(1-wz)^{-(\beta )}f(w)d\mu (w). \end{aligned}$$

The operator itself has been widely studied when $\mu$ is a positive Borel measure supported on the interval [0, 1). We study the boundedness of $K_{\mu ,1}$ acting on Hardy spaces and the boundedness of $K_{\mu ,\alpha }$, $\alpha >1$ acting on weighted Bergman spaces. Then we raise and answer some questions about the boundedness of those operators. Also, we find some special measures $\mu 's$ such that s-Hankel measure is equal to s-Carleson measure.