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Restricted Mean Value Property with non-tangential boundary behavior on Riemannian manifolds

Utsav Dewan

Abstract

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 $\mathbb{R}^2$.

Restricted Mean Value Property with non-tangential boundary behavior on Riemannian manifolds

Abstract

A well studied classical problem is the harmonicity of functions satisfying the restricted mean-value property (RMVP) for domains in . 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 .
Paper Structure (5 sections, 7 theorems, 57 equations)

This paper contains 5 sections, 7 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of two auxiliary functions
  4. An $L^\infty$ maximum principle
  5. Proofs of the main results

Key Result

Theorem 1.3

For a fixed $\alpha \in (0,\infty)$, we fix a constant $\kappa \in \left(0,\frac{\alpha}{4+\alpha}\right)$. If $u: \Omega \subset M \to \mathbb R$ is bounded, continuous, satisfies the Restricted Mean Value Property at each point $z \in \Omega$ on a geodesic sphere of radius $\rho(z) \le \kappa d(z, whenever $z$ approaches to $\xi$ through the non-tangential cone $\Gamma_\alpha(\xi)$, for almost e

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Definition 4.1
  • ...and 9 more