Mathematics

 Let $\Omega$ be a symmetric cone in $\mathbb{R}$ n and $T_\Omega = \mathbb{R}^n + i\Omega$, the tube domain over $\Omega$. Let $H^p (T_\Omega)$ be the Hardy space on $T_\Omega$ which is a higher dimension generalization of the classical Hardy space on the upper half plane. We consider the Carleson measure problem for Hardy space on $T_\Omega$. That is the problem of characterizing positive measures $\mu$ in $T_\Omega$ such that $H^p (T_\Omega)$ continuously imbedded into $L^q (T_\Omega, \mu)$. In this talk, I will sketch the solution of this problem in dimension one, that is the case of Hardy space on the upper half plane, given by L. Carleson (1962) for $p = q$, and $P$. Duren (1969) for $p < q$. I will also report on recent advances on this problem based on joint work with D. Bekolle and B. Sehba.