Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields |
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Authors: | Grillo Gabriele |
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Institution: | (1) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy |
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Abstract: | Let {X
i
}, i=1,...,m be a system of locally Lipschitz vector fields on DR
n
, such that the corresponding intrinsic metric 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 x
0 is sufficiently large. Then weighted Sobolev–Poincaré inequalities with weights given by power of (,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 Schrödinger operators H=–L+V are given. |
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Keywords: | Hardy inequality Rellich inequality Sobolev– Okikiolu inequality degenerate elliptic operators |
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