Green's Functions and Energy Decay on Homogeneous Spaces

TL;DR

This paper establishes Green's function estimates and energy decay principles on homogeneous spaces with Dirichlet forms, under Poincaré inequalities, extending classical potential theory to these geometric settings.

Contribution

It provides new Green's function bounds and energy decay results for homogeneous spaces satisfying Poincaré inequalities, generalizing classical analysis to these structures.

Findings

Green's function bounds are derived under Poincaré inequalities.

Energy decay principles analogous to Saint-Venant are established.

Results apply to a broad class of homogeneous spaces with Dirichlet forms.

Abstract

We consider a homogeneous space X=(X,d,m) of dimension $\nu \geq 1$ and a local regular Dirichlet form in L^{2}(X,m). We prove that if a Poincar\'{e} inequality holds on every pseudo-ball B(x,R) of X, with local characteristic constant c_{0}(x) and c_{1}(r), then a Green's function estimate from above and below is obtained. A Saint-Venant-like principle is recovered in terms of the Energy's decay.