Improved hardy inequalities on Riemannian manifolds

Abstract

We study the following version of Hardy-type inequality on a domain Ω in a Riemannian manifold (M,g): ∫Ω|∇u|pgραdVg≥(|p−1+β|p)p∫Ω|u|p|∇ρ|pg|ρ|pραdVg+∫ΩV|u|pραdVg,∀u∈C∞c(Ω). We provide sufficient conditions on p,α,β,ρ and V for which the above inequality holds. This generalizes earlier well-known works on Hardy inequalities on Riemannian manifolds. The functional setup covers a wide variety of particular cases, which are discussed briefly: for example, RN with p