C^n空间中超球外的调和函数的展式与超球上的解析函数Dirichlet边值问题

THE EXPANSION OF HARMONIC FUNCTION OUT OF THE UNIT BALL AND THE DIRICHLET PROBLEM FOR ANALYTIC FUNCTION ON THE UNIT BALL IN C￣n

本文构造了Ｃｎ空间中超球外的调和函数的展式，由此获得当ｎ≥２时，超球外解析函数或多重调和函数ｖ（ｚ），若满足｜ｖ（ｚ）｜＝ｏ（１），ｚ→∞，则ｖ（ｚ）＝０；给出了超球上的解析函数的Ｄｉｒｉｃｈｌｅｔ边值问题可解的充要条件．

In this paper, the expansion is obtained for harmonic function out of the unit ball in Cn. Using the result, we show that when n≥ 2, if u(z) is a analytic function or a pluriharmonic function out of the ball, if |u(z)|= o(1), z →∞, then u(z) ≡ 0, give the necessary and sufficient condition that the Dirichlet problem for theanalytic function on the unit ball is solvable.