Hardy inequalities with non standard remainder terms.

Starting point in this paper is the classical Hardy inequality with optimal constant and the lackness of extremals for funtions in the Sobolev space W^{1,p}(R^n). The lack of extremal has inspired improved versions, where the whole space R^n is replaced by any open bounded subset containing the origin and that amount to extra terms on the left-hand side that either involve integrals of |u|^p with weights depending on |x| which are less singular than |x|^{-p} at zero or integrals of |Du|^q with q<p. In this paper it is estabilished a stregthened version in the whole of R^n with a remainder term having a different nature. Such a remainder depends on a distance of u, in a suitable norm, from the family of those functions that can be regarded as the virtual extremals in the Hardy inequality.