BMO- and VMO-spaces of slice hyperholomorphic functions

TL;DR

This paper extends the study of Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball, focusing on BMO and VMO spaces, their properties, and relations to classical function spaces.

Contribution

It provides a detailed analysis of BMO and VMO spaces in the slice hyperholomorphic setting, including conformal invariance, Carleson measure characterizations, and duality relations.

Findings

Characterization of BMO and VMO spaces via Carleson measures

Relations established between these spaces and classical function spaces

Analysis of conformal invariance properties

Abstract

In this paper we continue the study of important Banach spaces of slice hyperholomorphic functions on the quaternionic unit ball by investigating the BMO- and VMO-spaces of slice hyperholomorphic functions. We discuss in particular conformal invariance and a refined characterization of these spaces in terms of Carleson measures. Finally we show the relations with the Bloch and Dirichlet space and the duality relation with the Hardy space $\mathcal{H}^1(\mathbb{D})$. The importance of these spaces in the classical theory is well known. It is therefore worthwhile to study their slice hyperholomorphic counterparts, in particular because slice hyperholomorphic functions were found to have several applications in operator theory and Schur analysis.