The fractional Hardy inequality with a remainder term

TL;DR

This paper derives an optimal fractional Hardy inequality with a remainder term for the fractional Laplacian on intervals and extends it to general domains, improving understanding of fractional operators and inequalities.

Contribution

It introduces a new Hardy inequality with an optimal constant and a lower-order remainder term for fractional Laplacians on various domains.

Findings

Derived the regional fractional Laplacian on power functions.

Proved Hardy inequality with an optimal constant and a remainder term.

Extended results to general domains using a new method.

Abstract

We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]