Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold

Abstract

We investigate the sharp constant for weighted fractional Hardy inequalities with the singularity on a flat submanifold of codimension $k$, where $1\leq k<d$. We also prove a weighted fractional Hardy inequality with a remainder. Using this result, we extend and derive a weighted version of the fractional Hardy-Sobolev-Maz'ya inequality with singularities on a flat submanifold. Furthermore, we obtain a weighted logarithmic fractional Hardy-Sobolev-Maz'ya inequality in the case of a singularity at the origin and we show that in this case, the fractional Hardy-Sobolev-Maz'ya inequality does not hold.