Integral representations of a class of harmonic functions in the half space

作者：Zhang, Yan Hui^*; Deng, Guan Tie; Qian, Tao

来源：Journal of Differential Equations, 2016, 260(2): 923-936.

DOI：10.1016/j.jde.2015.09.032

• Integral representation
• Positive part
• Modified Poisson kernel

In this article, motivated by the classic Hadamard factorization theorem about an entire function of finite order in the complex plane, we firstly prove that a harmonic function whose positive part satisfies some growth conditions, can be represented by its integral on the boundary of the half space. By using Nevanlinna's representation of harmonic functions and the modified Poisson kernel of the half space, we further prove a representation formula through integration against a certain measure on the boundary hyperplane for harmonic functions not necessarily continuous on the boundary hyperplane whose positive parts satisfy weaker growing conditions than the first question. The result is further generalized by involving a parameter m dealing with the singularity at the infinity.

• 单位
  北京师范大学; 北京工商大学