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 $A^{p}_{\lambda,w}({\mathbb{D}})$, 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 $\lambda\ge0$, 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 $2\lambda/(2\lambda+1)\le p\le1$, the boundedness of an operator from $A^{p}_{\lambda,w}({\mathbb{D}})$ 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 $A^{p}_{\lambda,w}({\mathbb{D}})$ for general weights $w$, and characterize the dual space of $A^{p}_{\lambda,w}({\mathbb{D}})$ 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 $A^{p}_{\lambda,w}({\mathbb{D}})$.