修改的Poisson核和调和函数的积分表示

Modified Poisson Kernel and Integral Representation of Harmonic Functions in Half-Plane

在本文中,对于半平面中的调和函数u(z),利用半平面中修改的Poisson核,证明了如果它的正部u^+(z)=max{u(z),0}满足某些限制增长条件,则它可以用半平面边界上的积分表示出来,并且它的负部u^-(z)=max{-u(z),0}也被类似的增长条件所控制,这一结果改进了在半平面中调和函数的某些经典结果。

In this paper, 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 in the boundary of the half plane and that its negative part u^-（z） = max{-u（z）, 0} can be dominated by a similar slowly growing condition. This improves some classical results about harmonic functions in a half-plane.