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

Abstract

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 $\Omega ⫋{\mathbb{R}}^n$, $n\ge 2$, be an arbitrary domain, $1<p<\infty$ and $\alpha +p>n$. Let ${\mathrm{d}}_{\Omega }(x)=\mathrm{dist}(x,\partial \Omega )$ denote the distance of a point $x\in \Omega$ to $\partial \Omega$. Then the following Hardy-type inequality holds $${\int }_{\Omega }\frac{|\nabla \phi {|}^p}{{\mathrm{d}}_{\Omega }^{\alpha }}\mathrm{d}x\ge {\left(\frac{\alpha +p-n}{p}\right)}^p{\int }_{\Omega }\frac{|\phi {|}^p}{{\mathrm{d}}_{\Omega }^{p+\alpha }}\mathrm{d}x\forall \phi \in C_c^{\infty }(\Omega ),$$ and the lower bound constant ${\left(\frac{\alpha +p-n}{p}\right)}^p$ is sharp.