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

Abstract

For $\lambda\ge0$, a $C^2$ function $f$ defined on the unit disk ${{\mathbb D}}$ is said to be $\lambda$-analytic if $D_{\bar{z}}f=0$, where $D_{\bar{z}}$ is the (complex) Dunkl operator given by $D_{\bar{z}}f=\partial_{\bar{z}}f-\lambda(f(z)-f(\bar{z}))/(z-\bar{z})$. The aim of the paper is to study several problems on the associated Bergman spaces $A^{p}_{\lambda}({{\mathbb D}})$ and Hardy spaces $H_{\lambda}^p({{\mathbb D}})$ for $p\ge2\lambda/(2\lambda+1)$, such as boundedness of the Bergman projection, growth of functions, density, completeness, and the dual spaces of $A^{p}_{\lambda}({{\mathbb D}})$ and $H_{\lambda}^p({{\mathbb D}})$, and characterization and interpolation of $A^{p}_{\lambda}({{\mathbb D}})$.