Weighted Hardy inequality with higher dimensional singularity on the boundary

Let $\Omega$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $\Sigma_k$ be a closed smooth submanifold of $\delta \Omega$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $\Sigma_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.