Hardy and Rellich Type Inequalities with Remainders

Abstract

Hardy and Rellich type inequalities with an additional term are proved for compactly supported smooth functions on open subsets of the Euclidean space. We obtain one-dimensional Hardy type inequalities and their multidimensional analogues in convex domains with the finite inradius. We use Bessel functions and the Lamb constant. The statements proved are a generalization for the case of arbitrary p ⩾ 2 of the corresponding inequality proved by F. G. Avkhadiev, K.-J. Wirths (2011) for p = 2. Also we establish Rellich type inequalities on arbitrary domains, regular sets, on domains with θ-cone condition and on convex domains.