Sharp extensions of analytic function spaces in the unit disk and related domains

In this expository paper we collect many recent advances in analytic function spaces of several complex variables related with trace problem. We consider various function space of analytic functions of several variables in various domains in Cn and provide or complete descriptions of traces or estimates of traces of various analytic function spaces in various domains obtained in recent years, by various authors. The problem to find sharp estimates of Hardy analytic function spaces in the unit polydisk first was posed by W. Rudin in 1969. Since then many papers appeared in literature. We collect in this expository paper not only already many known results on traces of various analytic function spaces in product domains but also discuss various new interesting results related with this problem. Related to trace problem various results were provided previously by G. Henkin, E. Amar, H. Alexander and various other authors. Finally note that our trace theorems are closely related with the Bergman type projections acting between function spaces with different dimensions.