Harnack Inequality on Homogeneous Spaces

We consider a connected, locally compact topological space X. We suppose that a pseudo-distance d is defined on X that is, d : X × X 7−→ R+ such that d (x, y) > 0 if and only if x 6= y; d (x, y) = d (y, x) ; d (x, z) ≤ γ [d (x, y) + d (y, z)] for all x, y, z ∈ X, where γ ≥ 1 is some given constant and we suppose that the pseudo-balls B (x, r) = {y ∈ X : d (x, y) < r} , r > 0, form a basis of open neighborhoods of x ∈ X. Moreover, we suppose that a (positive) Radon measure m is given on X, with suppm = X. The triple (X, d,m) is assumed to satisfy the following property: There exist some constants 0 < R0 ≤ +∞, ν > 0 and c0 > 0, such that