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

TL;DR

This paper investigates the boundedness of operators on weighted Bergman spaces linked to a class of generalized analytic functions called $ ext{lambda}$-analytic functions, providing new criteria and characterizations for these operators.

Contribution

It introduces new boundedness criteria for operators on $ ext{lambda}$-analytic Bergman spaces, including conditions for sequence multipliers, dual space characterization, and Carleson-type boundedness.

Findings

Boundedness depends on a single vector-valued $ ext{lambda}$-analytic function.

Characterization of dual spaces for specific weights.

Necessary and sufficient conditions for sequence multipliers and Carleson measures.

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 $A^{p}_{\lambda,w}({\mathbb{D}})$.