Abstract Assume that p ∈ ( 1 , ∞ ] and u = P h [ ϕ ] , where ϕ ∈ L p ( S n − 1 , R n ) . Then for any x ∈ B n , we obtain the sharp inequalities | u ( x ) | ≤ C q 1 q ( x ) ( 1 − | x | 2 ) n − 1 p ‖ ϕ ‖ L p and | u ( x ) | ≤ C q 1 q ( 1 − | x | 2 ) n − 1 p ‖ ϕ ‖ L p for some function C q ( x ) and constant C q in terms of Gauss hypergeometric and Gamma functions, where q is the conjugate of p . This result generalizes and extends some known results from harmonic mapping theory (Kalaj and Markovic (2012, Theorems 1.1 and 1.2) and Axler et al. (1992, Proposition 6.16)). The proofs are mainly based on certain characterizations of the radial eigenfunctions of the hyperbolic Laplacian Δ h , which are of independent interest.

中文翻译：

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Lp 中函数的双曲泊松积分的最优估计，其中 p>1 和双曲拉普拉斯算子的径向本征函数

摘要 假设 p ∈ ( 1 , ∞ ] 且 u = P h [ ϕ ] , 其中 ϕ ∈ L p ( S n − 1 , R n ) . 那么对于任意 x ∈ B n ，我们得到尖锐的不等式 | u ( x ) | ≤ C q 1 q ( x ) ( 1 − | x | 2 ) n − 1 p ‖ ϕ ‖ L p 和 | u ( x ) | ≤ C q 1 q ( 1 − | x | 2 ) n − 1 p ‖ ϕ ‖ L p 对于某些函数 C q ( x ) 和常数 C q 根据高斯超几何和 Gamma 函数计算，其中 q 是 p 的共轭。这个结果概括和扩展了调和映射理论 (Kalaj) 的一些已知结果和 Markovic (2012, Theorems 1.1 and 1.2) and Axler et al. (1992, Proposition 6.16)). 证明主要基于双曲拉普拉斯算子 Δ h 的径向本征函数的某些特征，它们是独立的。