Some problems related with holomorphic functions on Tube Domains over Light Cones

In this survey, we consider two kinds of problems on tube domains over light cones. The first one is related with Poisson-Szeg\"{o} integrals $F$. When $F$ is a bounded real function and satisfies an appropriate smoothness up to the boundary, then $F$ is necessarily the real part of a holomorphic function. The second one is the $L^p$ boundedness of the Bergman projection, for which known positive and negative results leave a gap between them. This gives an illustration of activities within HARP.