On weighted $L^p$-Hardy inequality on domains in ${\mathbb{R}}^n$

We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result of F.~G.~Avkhadiev (2006) Theorem: Let $\Omega \subsetneqq $\mathbb{R}$^n$,$n\geq 2$,beanarbitrarydomain,$1<p<\infty$and$\alpha + p>n$\mathrm{.}Let$$\mathrm{d}$_\Omega(x) =$\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}$(x,\partial \Omega )$ denote the distance of a point $x\in \mathrm{\Omega }$ to $\mathrm{\partial }\mathrm{\Omega }$. Then the following Hardy-type inequality holds $${\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }\mathrm{\nabla }\phi {\mathrm{\mid }}^p}{{\mathrm{d}}_{\mathrm{\Omega }}^{\alpha }}\mathrm{d}x\ge {\left(\frac{\alpha +p-n}{p}\right)}^p{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }\phi {\mathrm{\mid }}^p}{{\mathrm{d}}_{\mathrm{\Omega }}^{p+\alpha }}\mathrm{d}x\mathrm{\forall }\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega }),$$ and the lower bound constant $\left( \frac{\alpha +p-n}{p}\right)^p$ is sharp.