The Duren-Carleson theorem in tube domains over symmetric cones

TL;DR

This paper extends the Duren-Carleson theorem to tube domains over symmetric cones, characterizing measures for Hardy space embeddings and multipliers, thus generalizing classical results to a broader geometric setting.

Contribution

It generalizes the Duren-Carleson theorem to tube domains over symmetric cones and characterizes measures and multipliers in this context.

Findings

Characterization of measures for Hardy space embeddings.

Extension of classical results to symmetric cone domains.

Complete description of multipliers from Hardy to Bergman spaces.

Abstract

In the setting of tube domains over symmetric cones, $T_\Omega$, we study the characterization of the positive Borel measures $\mu$ for which the Hardy space $H^p$ is continuously embedded into the Lebesgue space $L^q (T_\Omega, d\mu)$, $0<p<q<\infty.$ Extending a result due to Blasco for the unit disc, we reduce the problem to standard measures. We obtain that a Hardy space $H^{p}$, $1\leq p < \infty,$ embeds continuously in weighted Bergman spaces with larger exponents. Finally we use this result to characterize multipliers from $H^{2m}$ to Bergman spaces for every positive integer $m$.