Carleson-type embeddings with closed range

TL;DR

This paper characterizes Carleson measures for Hardy and Bergman spaces that produce closed-range embedding operators, extending to a more general $(p,q)$-Carleson measure setting.

Contribution

It provides a new characterization of measures ensuring closed-range embeddings for Hardy and Bergman spaces, including a broader $(p,q)$-Carleson measure framework.

Findings

Characterization of Carleson measures with closed range embeddings in Hardy spaces.

Extension of results to $(p,q)$-Carleson measures.

Solution of a similar problem in Bergman spaces.

Abstract

We characterize the Carleson measures $\mu$ on the unit disk for which the image of the Hardy space $H^p$ under the corresponding embedding operator is closed in $L^p(\mu)$. In fact, a more general result involving $(p,q)$-Carleson measures is obtained. A similar problem is solved in the setting of Bergman spaces.