Sobolev Inequalities on Homogeneous Spaces

Abstract

We consider a homogeneous space X = (X, d, m) of dimension ν ≥ 1 and a local regular Dirichlet form a in L ²(X, m). We prove that if a Poincaré inequality of exponent 1 ≤ p < ν holds on every pseudo-ball B(x, R) of X, then Sobolev and Nash inequalities of any exponent q ∈ [p, ν), as well as Poincaré inequalities of any exponent q ∈ [p, +∞), also hold on B(x, R).