Quaternionic Carleson measures

TL;DR

This paper introduces a unified framework for quaternionic Banach spaces of slice regular functions and characterizes Carleson measures within these spaces based on their holomorphic counterparts.

Contribution

It provides a general construction of quaternionic Banach spaces from holomorphic spaces and characterizes Carleson measures in this quaternionic setting.

Findings

Unified approach to Carleson measures in quaternionic spaces

Construction encompasses all known quaternionic Banach spaces

Characterization reduces to holomorphic Carleson measures

Abstract

In this paper we provide a general construction of a quaternionic Banach space of slice regular functions from a given Banach space of holomorphic functions, which we call its quaternionic lift. To the best of our knowledge, this construction encompasses all known examples of quaternionic Banach spaces of slice regular functions in the literature. Our main result is a characterization of Carleson and vanishing Carleson measures for such quaternionic Banach function spaces in terms of the corresponding Carleson measures of the underlying holomorphic function space. This offers a unified approach to a problem that so far has been treated on a case-by-case basis.