Coarse and Precise $L^p$-Green Potential Estimates on Noncompact Riemannian Manifolds

TL;DR

This paper develops coarse and precise estimates for Green's functions on noncompact Riemannian manifolds, utilizing Sobolev, Nash, and isoperimetric inequalities to analyze their asymptotic behavior and related functionals.

Contribution

It introduces new a priori Green's function estimates on noncompact manifolds combining multiple inequalities and asymptotic analysis methods.

Findings

Derived sharp Green's function bounds using isoperimetric inequalities.

Evaluated the asymptotic behavior of Green's function at infinity.

Connected Green's function estimates with harmonic radius and Ricci curvature.

Abstract

We are concerned about the coarse and precise aspects of a priori estimates for Green's function of a regular domain for the Laplacian-Betrami operator on any $3\le n$-dimensional complete non-compact boundary-free Riemannian manifold through the square Sobolev/Nash/logarithmic-Sobolev inequalities plus the rough and sharp Euclidean isoperimetric inequalities. Consequently, we are led to evaluate the critical limit of an induced monotone Green's functional using the asymptotic behavior of the Lorentz norm deficit of Green's function at the infinity, as well as the harmonic radius of a regular domain in the Riemannian manifold with nonnegative Ricci curvature.