Optimal Lᵖ Hardy-type inequalities

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 satisfying Q(φ ) ≥ ∫ ΩW|φ |ᵖ\:dν \:∀ 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 by W: = (p−1/p)ᵖ|∇G/G|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.