One dimensional Weighted Hardy's Inequalities and application

TL;DR

This paper improves one-dimensional weighted Hardy inequalities with sharp remainders and extends these results to n-dimensional domains, with applications to variational problems.

Contribution

It introduces sharper weighted Hardy inequalities with remainders and generalizes them to higher dimensions for bounded smooth domains.

Findings

Improved one-dimensional weighted Hardy inequalities with sharp remainders.

Extended Hardy inequalities to n-dimensional bounded smooth domains.

Applicable to variational problems in future research.

Abstract

In the present paper we shall improve one dimensional weighted Hardy inequalities with one-sided boundary condition by adding sharp remainders. As an application, we shall establish n dimensional weighted Hardy inequalities in a bounded smooth domain with weight functions being powers of the distance function d(x) to the boundary. Our results will be applicable to variational problems in a coming paper.