Hardy’s inequality for 𝑊^{1,𝑝}₀-functions on Riemannian manifolds

Abstract

We prove that for every Riemannian manifold X \mathcal {X} with the isoperimetric profile of particular type there holds an inequality of Hardy type for functions of the class W 0 1 , p ( X ) W_0^{1,p}( \mathcal {X}) . We also study manifolds satisfying Hardy’s inequality and, in particular, we establish an estimate for the rate of growth of the weighted volume of the noncompact part of such a manifold.