Multidimensional Hardy Type Inequalities with Remainders

Abstract

Hardy type inequalities with an additional nonnegative-term are established for compactly supported smooth functions on arbitrary open subsets and on convex domains of the Euclidean space. We prove Hardy-type inequalities in spatial domains with finite inner radius. Weight functions depend on the distance function to the boundary of the domain. We obtain one-dimensional L₁-inequalities. In particular cases we obtained sharp constants. Also new Hardy type inequality with remainders for the Riemann-Liouville fractional integrals is proved.