On improved Hardy-type inequalities in Log-Weighted Sobolev Spaces

Abstract

We prove sharp Hardy-type inequalities in Log-weighted Sobolev spaces. Due to the non-attainability of the sharp constant, we improve these inequalities by adding suitable remainder terms consisting of optimal weights and sharp constants. Consequently, this gives an infinite series improvement in Log-weighted Sobolev spaces. Next, by using these improved Hardy-type inequalities, we also prove an optimal Leray-Trudinger-type improvement. Furthermore, we derive interpolation between Hardy-type inequalities and Leray-Trudinger-type inequalities.