An optimal Hardy-Morrey inequality

TL;DR

This paper enhances the Hardy inequality for the case p>n by incorporating an optimal weighted Hölder semi-norm, providing local improvements and refining related Sobolev and Hardy inequalities with the best constants.

Contribution

It introduces an optimal weighted Hölder semi-norm to improve the Hardy inequality for p>n, including local enhancements and refined inequalities with optimal constants.

Findings

Improved Hardy inequality with optimal weighted Hölder semi-norm

Local improvement of the Hardy inequality

Refinement of Sobolev and Hardy inequalities with best constants

Abstract

In this work we improve the sharp Hardy inequality in the case $p>n$ by adding an optimal weighted Hoelder semi-norm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for $p>n$ and the Hardy inequality, the latter having the best constant.