Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball

TL;DR

This paper characterizes Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball, using geometric conditions involving pseudohyperbolic discs and balls, highlighting differences from complex cases.

Contribution

It provides a new geometric characterization of Carleson measures in quaternionic spaces, extending classical complex analysis results to quaternionic Hardy and Bergman spaces.

Findings

Characterization of slice Carleson measures for quaternionic Hardy and Bergman spaces.

Use of axially symmetric completion of pseudohyperbolic discs for Bergman spaces.

Demonstration that pseudohyperbolic balls do not characterize Carleson measures.

Abstract

We study a characterization of slice Carleson measures and of Carleson measures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces $\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc in a complex plane. We also show that a characterization in terms of pseudohyperbolic balls is not possible.