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 $Ω\subsetneqq \mathbb{R}^n$, $n\geq 2$, be an arbitrary domain, $1<p<\infty$ and $α+ p>n$. Let $\mathrm{d}_Ω(x) =\mathrm{dist}(x,\partial Ω)$ denote the distance of a point $x\in Ω$ to $\partial Ω$. Then the following Hardy-type inequality holds $\int_{Ω}\frac{|\nabla φ|^p}{\mathrm{d}_Ω^α}\,\mathrm{d}x \geq \left( \frac{α+p-n}{p}\right)^p \int_{Ω}\frac{|φ|^p}{\mathrm{d}_Ω^{p+α}}\,\mathrm{d}x \qquad \forall φ\in C^{\infty }_c(Ω),$ and the lower bound constant $\left( \frac{α+p-n}{p}\right)^p$ is sharp.

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