Weighted fractional Hardy inequalities with singularity on any flat submanifold

TL;DR

This paper generalizes weighted fractional Hardy inequalities with singularities from points and half-spaces to any flat submanifold of codimension k, including the critical case with logarithmic weights.

Contribution

It extends previous results to arbitrary flat submanifolds of codimension k and addresses the critical case with logarithmic weights.

Findings

Established weighted fractional Hardy inequalities for singularities on any flat submanifold.

Derived inequalities for the critical case with logarithmic weights.

Generalized previous results to broader geometric settings.

Abstract

We extend the work of Dyda and Kijaczko by establishing the corresponding weighted fractional Hardy inequalities with singularities on any flat submanifolds. While they derived weighted fractional Hardy inequalities with singularities at a point and on a half-space, we generalize these results to handle singularities on any flat submanifold of codimension $k$, where $1<k<d$. Furthermore, we also address the critical case $sp=k+\alpha+ \beta$ and establish weighted fractional Hardy inequality with appropriate logarithmic weight function.