Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields

Let Xi, i=1,m be a system of locally Lipschitz vector fields on D⊂Rn, such that the corresponding intrinsic metric ϱ 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 L1 Poincaré inequality holds for the vector fields at hand (thus including the case of Hörmander vector fields) and that the local homogeneous dimension near a point x0 is sufficiently large. Then weighted Sobolev–Poincaré inequalities with weights given by power of ϱ(⋅,x0) 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 Schrödinger operators H=−L+V are given.