黎曼流形上具有非切向边界行为的受限均值性质
Restricted Mean Value Property with non-tangential boundary behavior on Riemannian manifolds
AI总结：
本文在黎曼流形中具有光滑边界的预紧区域及纯指数体积增长的非正曲率调和流形上，用非切向边界极限替代无限制边界极限，证明了满足受限均值性质的函数的调和性，推广了 Fenton 的经典结果。
摘要：
一个被广泛研究的经典问题是：在 $\mathbb{R}^n$ 中区域上满足受限均值性质（RMVP）的函数的调和性。最近，作者与 Biswas 在黎曼流形的一般框架下研究了该问题，并得到了用函数在边界的一个全测度子集上的无限制边界极限来表述的结果。然而，在关于调和函数边界行为的经典 Fatou–Littlewood 型定理的背景下，一个自然的问题是：用更自然的非切向边界极限概念来替代无限制边界极限条件。本文的目的是在局部情形下，对黎曼流形中具有光滑边界的预紧区域，以及在全局情形下，对具有纯指数体积增长的非正曲率调和流形，回答这一问题。这推广了 Fenton 关于 $\mathbb{R}^2$ 中单位圆盘的经典结果。
英文摘要：
A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP) for domains in $\mathbb{R}^n$. Recently, the author along with Biswas investigated the problem in the general setting of Riemannian manifolds and obtained results in terms of unrestricted boundary limits of the function on a full measure subset of the boundary. However in the context of classical Fatou-Littlewood type theorems for the boundary behavior of harmonic functions, a genuine query is to replace the condition on unrestricted boundary limits with the more natural notion of non-tangential boundary limits. The aim of this article is to answer this question in the local setup for pre-compact domains with smooth boundary in Riemannian manifolds and in the global setup for non-positively curved Harmonic manifolds of purely exponential volume growth. This extends a classical result of Fenton for the unit disk in $\mathbb{R}^2$.
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