Weighted fractional Hardy inequalities with singularity on any flat submanifold

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.