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

Abstract

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