Analytic boundary value problems on classical domains

In this paper analytic boundary value problems for some classical domains in ℂn are developed by using the harmonic analysis due to L.K. Hua. First it is discussed for the version of one variable in order to induce the relation between the analytic boundary value problem and the decomposition of function space L2 on the boundary manifold. Then an easy example of several variables, the version of torus in ℂ2, is stated. For the noncommutative classical group LI, the characteristic boundary of a kind of bounded symmetric domain in ℂ4, the boundary behaviors of the Cauchy integral are obtained by using both the harmonic expansion and polar coordinate transformation. At last we obtain the conditions of solvability of Schwarz problem on LI, if so, the solution is given explicitly.