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

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