Boundedness of operators on the Bergman spaces associated with a class of generalized analytic functions

Abstract

The purpose of the paper is to study the operators on the weighted Bergman spaces on the unit disk $\mathbb{D}$, denoted by $A_{\lambda ,w}^p(\mathbb{D})$, that are associated with a class of generalized analytic functions, named the $\lambda$-analytic functions, and with a class of radial weight functions $w$. For $\lambda \ge 0$, a $C^2$ function $f$ on $\mathbb{D}$ is said to be $\lambda$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f={\partial }_{\bar{z}}f-\lambda (f(z)-f(\bar{z}))/(z-\bar{z})$. It is shown that, for $2\lambda /(2\lambda +1)\le p\le 1$, the boundedness of an operator from $A_{\lambda ,w}^p(\mathbb{D})$ into a Banach space depends only upon the norm estimate of a single vector-valued $\lambda$-analytic function. As applications, we obtain a necessary and sufficient conditions of sequence multipliers on the spaces $A_{\lambda ,w}^p(\mathbb{D})$ for general weights $w$, and characterize the dual space of $A_{\lambda ,w}^p(\mathbb{D})$ for the power weight $w=(1-|z{|}^2{)}^{\alpha -1}$ with $\alpha >0$, and also give a sufficient condition of Carleson type for boundedness of multiplication operators on $A_{\lambda ,w}^p(\mathbb{D})$.

42 pages