Fractional Hardy inequality with singularity on submanifold

TL;DR

This paper proves fractional Hardy inequalities on bounded domains with weights based on the inverse distance to the boundary, addressing critical and non-critical cases with optimal corrections.

Contribution

It establishes fractional Hardy inequalities involving singularities on submanifolds of various codimensions, including critical cases with logarithmic corrections.

Findings

Proved fractional Hardy inequalities with inverse distance weights.

Addressed critical case with optimal logarithmic corrections.

Extended inequalities to various codimensions and parameter ranges.

Abstract

We establish fractional Hardy inequality on bounded domains in $\mathbb{R}^{d}$ with inverse of distance function from smooth boundary of codimension $k$, where $k=2, \dots,d$, as weight function. The case $sp=k$ is the critical case, where optimal logarithmic corrections are required. All the other cases of $sp<k$ and $sp>k$ are also addressed.