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

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• 作者：Yan Hui Zhang^a ; ^b ; ^{zhangyanhui@th.btbu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; ^{yanhui.zhang@ucc.ie" class="auth_mail" title="E-mail the corresponding author} ; Guan Tie Deng^c ; ^{denggt@bnu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Tao Qian^d ; ^{fsttq@umac.mo" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Integral representation ; Positive part ; Modified Poisson kernel
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：260
• 期：2
• 页码：923-936
• 全文大小：244 K

文摘

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.