Harnack Inequality on Homogeneous Spaces

We consider a homogeneous space $X=(X,d,m)$ of dimension $\nu \ge 1$ and a local regular Dirichlet form in $L^2(X,m)\mathrm{.}$ 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 $c_1$