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

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