A strong form of Hardy Type Inequalities on Domains of the Euclidean Space

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