A Strong Form of Hardy Type Inequalities on Domains of the Euclidean Space

Abstract

We prove new integral inequalities for real-valued test functions defined on subdomains of the Euclidean space. Namely, we obtain several new Hardy-type inequalities that contain the scalar product of gradients of test functions and of the gradient of the distance function from the boundary of an open subset of the Euclidean space. Our method of proof is based on interior and exterior approximations of a given domain by sequences of simplest domains and has two important ingredients. The first one is approximations of a given domain by elementary domains that admit a special partition. The second ingredient is presented in this paper by Theorems 1 and 2 about convergence everywhere of the sequences of distance functions from boundary of approximating elementary domains as well as about convergence almost everywhere for their gradients. In the proofs we also use some basic theorems due to Rademacher, Hardy, Motzkin and Hadwiger.