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

TL;DR

This paper provides an alternative proof of a weighted $L^p$-Hardy inequality involving the distance to the boundary in domains of Euclidean space, establishing sharp constants using criticality theory.

Contribution

It offers a new proof of an existing Hardy inequality with sharp constants for arbitrary domains in $\,\mathbb{R}^n$ using criticality theory.

Findings

Proved the weighted Hardy inequality with sharp constant.

Extended the inequality to arbitrary domains in Euclidean space.

Used criticality theory as a novel proof technique.

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 $\Omega \subsetneqq \mathbb{R}^n$, $n\geq 2$, 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 $$ \int_{\Omega }\frac{|\nabla \varphi |^p}{\mathrm{d}_\Omega^{\alpha}}\,\mathrm{d}x \geq \left( \frac{\alpha +p-n}{p}\right)^p \int_{\Omega }\frac{|\varphi|^p}{\mathrm{d}_\Omega^{p+\alpha}}\,\mathrm{d}x \qquad \forall \varphi\in C^{\infty }_c(\Omega),$$ and the lower bound constant $\left( \frac{\alpha +p-n}{p}\right)^p$ is sharp.