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

TL;DR

This paper introduces a new Bloch type space associated with mbda-analytic functions defined via Dunkl operators, exploring its properties, characterizations, and bounded integral operators, with applications to duality and Bergman spaces.

Contribution

It defines the mbda-Bloch space for mbda-analytic functions, characterizes it using higher-order operators, and establishes boundedness of a general integral operator.

Findings

The mbda-Bloch space mbda{7}(4) is well-defined and its properties are established.

Functions in mbda{7}(4) are characterized via higher-order Dunkl operators.

A general integral operator is bounded from L^{\u221e}(4) onto mbda{7}(4).

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.