$\varepsilon$-approximability and Quantitative Fatou Theorem on Riemannian Manifolds

TL;DR

This paper extends the Quantitative Fatou Theorem and $ ext{ε}$-approximation techniques to Lipschitz domains on complete Riemannian manifolds, covering harmonic and A-harmonic functions, thereby generalizing prior Euclidean results.

Contribution

It introduces an $ ext{ε}$-approximation lemma for Riemannian manifolds and proves the Quantitative Fatou Theorem in this broader geometric setting.

Findings

Proves the Quantitative Fatou Theorem on Riemannian manifolds.

Extends $ ext{ε}$-approximation lemma to manifold setting.

Applies results to harmonic and A-harmonic functions.

Abstract

We prove the Quantitative Fatou Theorem for Lipschitz domains on complete Riemannian manifolds. This requires extending the $\varepsilon$-approximation lemma to the manifold setting. Our studies apply to harmonic functions, as well as to a class of A-harmonic functions on manifolds. The presented results extend works by Dahlberg, Garnett, Bortz and Hofmann.