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The purpose of the paper is to study the operators on the weighted Bergmanspaces on the unit disk ${\mathbb{D}}$, denoted by$A^{p}_{\lambda,w}({\mathbb{D}})$, that are associated with a class ofgeneralized analytic functions, named the $\lambda$-analytic functions, andwith 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 isshown that, for $2\lambda/(2\lambda+1)\le p\le1$, the boundedness of anoperator from $A^{p}_{\lambda,w}({\mathbb{D}})$ into a Banach space dependsonly upon the norm estimate of a single vector-valued $\lambda$-analyticfunction. As applications, we obtain a necessary and sufficient conditions ofsequence multipliers on the spaces $A^{p}_{\lambda,w}({\mathbb{D}})$ forgeneral 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 forboundedness of multiplication operators on $A^{p}_{\lambda,w}({\mathbb{D}})$.

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