Some aspects of the Bergman and Hardy spaces associated with a class of generalized analytic functions

Abstract

For $λ\ge0$, a $C^2$ function $f$ defined on the unit disk ${\mathbb D}$ is said to be $λ$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f=\partial_{\bar{z}}f-λ(f(z)-f(\bar{z}))/(z-\bar{z})$. The aim of the paper is to study several problems on the associated Bergman spaces $A^{p}_λ({\mathbb D})$ and Hardy spaces $H_λ^p({\mathbb D})$ for $p\ge2λ/(2λ+1)$, such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of $A^{p}_λ({\mathbb D})$ and $H_λ^p({\mathbb D})$, and characterization and interpolation of $A^{p}_λ({\mathbb D})$.