A Bloch type space associated with λ-analytic functions

Abstract

For $\hat{I}»\ge 0$, the so-called $\hat{I}»$-analytic functions are defined in terms of the (complex) Dunkl operators $D_z$ and $D_{\bar{z}}$. In the paper we introduce a Bloch type space on the disk $\mathbb{D}$ associated with $\hat{I}»$-analytic functions, called the $\hat{I}»$-Bloch space and denoted by ${\mathfrak{B}}_{\hat{I}}»(\mathbb{D})$. Various properties of the $\hat{I}»$-Bloch space ${\mathfrak{B}}_{\hat{I}}»(\mathbb{D})$ are proved. We give a characterization of functions in ${\mathfrak{B}}_{\hat{I}}»(\mathbb{D})$ by means of the higher-order operators $(D_z\circ z{)}^n$ for $n\ge 2$. A general integral operator is proved to be bounded from $L^{\infty }(\mathbb{D})$ onto ${\mathfrak{B}}_{\hat{I}}»(\mathbb{D})$, and as an application, the dual relation of ${\mathfrak{B}}_{\hat{I}}»(\mathbb{D})$ and the $\hat{I}»$-Bergman space ($p=1$) is verified.

18 pages