The Schwarz problem in a Siegel domain

摘要

In this article we consider the Schwarz problem for holomorphic functions in the Siegel domain D, the boundary of which is the Heisenberg group H. We first prove the Cauchy—Szego formula for D by using the same formula for the unit ball. Then the Plemelj formula for H is proved. At last, we get solvability conditions for the Schwarz problem, and under these conditions its general solution is given by an explicit formula.