半平面中调和函数的积分表示

Integral Representation of Harmonic Function in a Half-Plane

中文关键词:  修改的Poisson核  调和函数  积分表示.

英文关键词:integral representation  harmonic functions  modified Poisson kernel.

中文摘要:

      对于右半平面中的调和函数,利用修改的Poisson核,证明了如果它的正部满足一定的增长条件,则可以把它用 半平面边界上的测度的积分表示出来.

英文摘要:

      In this article, we consider the integral representation of harmonic functions. Using a property of the modified Poisson kernel in a half plane, we prove that a harmonic function $u(z)$ in a half plane with its positive part $u^ (z)=\max\{u(z),0\}$ satisfying a slowly growing condition can be represented by its integral of a measure on the boundary of the half plan.