A Bloch type space associated with λ-analytic functions

Abstract

For $λ\ge0$, the so-called $λ$-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 $λ$-analytic functions, called the $λ$-Bloch space and denoted by ${\mathfrak{B}}_λ({\mathbb D})$. Various properties of the $λ$-Bloch space ${\mathfrak{B}}_λ({\mathbb D})$ are proved. We give a characterization of functions in ${\mathfrak{B}}_λ({\mathbb D})$ 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 ${\mathfrak{B}}_λ({\mathbb D})$, and as an application, the dual relation of ${\mathfrak{B}}_λ({\mathbb D})$ and the $λ$-Bergman space ($p=1$) is verified.

Theorems & Definitions (19)

• Definition 1.1
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Theorem 2.4
• Lemma 3.1
• Proposition 3.2
• Proposition 3.3
• Lemma 3.4
• Lemma 3.5