Green's Functions and Energy Decay on Homogeneous Spaces

Abstract

We consider a homogeneous space X=(X,d,m) of dimension $ν\geq 1$ and a local regular Dirichlet form in L^{2}(X,m). We prove that if a Poincaré 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.