Harnack Inequality on Homogeneous Spaces

TL;DR

This paper proves that on homogeneous spaces, a Poincaré inequality implies Harnack's inequality for local regular Dirichlet forms, establishing a key link between geometric and analytic properties.

Contribution

It demonstrates that Poincaré inequalities on homogeneous spaces lead to Harnack inequalities for Dirichlet forms, extending classical results to a broader geometric setting.

Findings

Harnack's inequality holds under Poincaré inequality assumptions

The result applies to spaces with homogeneous structure

Constants depend on local geometric properties

Abstract

We consider a homogeneous space $X=(X,d,m) $ of dimension $\nu\geq1$ 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 $X$, then an Harnack's inequality can be proved on the same ball with local characteristic constant $c_{0}$ and $c_{1}$