Optimal $L^p$ Hardy inequalities

TL;DR

This paper investigates the optimal Hardy inequalities in the context of $L^p$ spaces, aiming to find the largest possible nonnegative weights that satisfy certain integral inequalities on punctured domains.

Contribution

It introduces a method to determine the maximal Hardy-type weights for $L^p$ inequalities, extending the understanding of Hardy inequalities in punctured domains.

Findings

Identifies the largest Hardy weights for given $L^p$ settings.

Provides a framework for optimal Hardy inequalities on punctured domains.

Establishes conditions under which these inequalities hold.

Abstract

Let $\mathcal{Q}(\varphi):=\int_\Omega \big(|\nabla \varphi|^p+V|\varphi|^p\big)\dnu$ on $\core$, and assume that $\mathcal{Q}\geq 0$. The aim of the paper is to obtain ''as large as possible" nonnegative (optimal) Hardy-type weight $W$ satisfying $$\mathcal{Q}(\varphi)\geq \int_{\Omega} W|\varphi|^p\dnu \quad\forall \varphi\in\core,$$ on punctured domains $\Omega$.