Optimal Lp Hardy-type inequalities

Abstract

Let Ω be a domain in Rn or a noncompact Riemannian manifold of dimension n≥2, and 1<p<∞. Consider the functional Q(φ):=∫Ω(|∇φ|p+V|φ|p)dν defined on C0∞(Ω), and assume that Q≥0. The aim of the paper is to generalize to the quasilinear case (p≠2) some of the results obtained in [6] for the linear case (p=2), and in particular, to obtain “as large as possible” nonnegative (optimal) Hardy-type weight W satisfyingQ(φ)≥∫ΩW|φ|pdν∀φ∈C0∞(Ω).Our main results deal with the case where V=0, and Ω is a general punctured domain (for V≠0 we obtain only some partial results). In the case 1<p≤n, an optimal Hardy-weight is given byW:=(p−1p)p|∇GG|p, where G is the associated positive minimal Green function with a pole at 0. On the other hand, for p>n, several cases should be considered, depending on the behavior of G at infinity in Ω. The results are extended to annular and exterior domains.