A Bloch type space associated with {\lambda}-analytic functions

Abstract

For $\lambda\ge0$, the so-called $\lambda$-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 $\lambda$-analytic functions, called the $\lambda$-Bloch space and denoted by ${\mathfrak{B}}_{\lambda}({\mathbb D})$. Various properties of the $\lambda$-Bloch space ${\mathfrak{B}}_{\lambda}({\mathbb D})$ are proved. We give a characterization of functions in ${\mathfrak{B}}_{\lambda}({\mathbb D})$ by means of the higher-order operators $(D_z\circ z)^n$ for $n\ge2$. A general integral operator is proved to be bounded from $L^{\infty}({\mathbb D})$ onto ${\mathfrak{B}}_{\lambda}({\mathbb D})$, and as an application, the dual relation of ${\mathfrak{B}}_{\lambda}({\mathbb D})$ and the $\lambda$-Bergman space ($p=1$) is verified.