Some Carleson measures for the Hilbert-Hardy space of tube domains over symmetric cones

TL;DR

This paper characterizes radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones and explores embedding properties of derivatives, reducing complex cases to known settings like upper-half planes.

Contribution

It provides a complete characterization of Carleson measures for these spaces and simplifies the analysis of derivatives by linking to classical upper-half plane problems.

Findings

Full characterization of radial Carleson measures for the space.

Criteria for measures ensuring continuity of embedding operators for derivatives.

Reduction of complex embedding problems to classical upper-half plane cases.

Abstract

In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding embedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two upper-half planes.