Hardy and Rellich-type inequalities for metrics defined by vector fields

Let {X-i}, i = 1,...,m be a system of locally Lipschitz vector fields on D subset of R-n,such that the corresponding intrinsic metric rho is well-defined and continuous w.r.t. the Euclidean topology. Suppose that the Lebesgue measure is doubling w.r.t. the intrinsic balls, that a scaled L-1 Poincare inequality holds for the vector fields at hand (thus including the case of Hormander vector fields) and that the local homogeneous dimension near a point x(0) is sufficiently large. Then weighted Sobolev-Poincare inequalities with weights given by power of rho(., x(0)) hold; as particular cases, they yield non-local analogues of both Hardy and Sobolev-Okikiolu inequalities. A general argument which shows how to deduce Rellich-type inequalities from Hardy inequalities is then given: this yields new Rellich inequalities on manifolds and even in the uniformly elliptic case. Finally, applications of Sobolev-Okikiolu inequalities to heat kernel estimates for degenerate subelliptic operators and to criteria for the absence of bound states for Schrodinger operators H = -L + V are given.