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

TL;DR

This paper investigates the properties of Bergman and Hardy spaces linked to a class of generalized analytic functions defined via Dunkl operators, focusing on boundedness, growth, density, duality, and interpolation.

Contribution

It introduces and analyzes new function spaces associated with Dunkl operators, extending classical analytic function theory to this generalized setting.

Findings

Boundedness of the Bergman projection established

Growth estimates for functions in the spaces derived

Density and duality properties characterized

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