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

In the setting of a general tube domain over a symmetric cone, we obtain a full characterization of measures of the form φ(y)dxdy\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (y)\,dxdy$$\end{document} which are Carleson measures for the Hilbert–Hardy space; for large derivatives, we also obtain a full characterization of general positive measures for which the corresponding embedding operator is continuous. Restricting to the case of the Lorentz cone of dimension three, we prove that by freezing one or two secondary 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.