The Duren-Carleson theorem in tube domains over symmetric cones

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 $H^p$, $1\le p<\infty ,$ 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 $m$.

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