On a Hardy-Morrey inequality

TL;DR

This paper develops a foundational theory for Hardy-Morrey inequalities, exploring their geometric implications, sharp constants, and conditions for extremal functions, extending classical Morrey and Hardy inequalities.

Contribution

It introduces a basic theoretical framework for Hardy-Morrey inequalities, analyzing their sharp constants and geometric dependencies, and studies extremal functions saturating the inequality.

Findings

Established the relationship between domain geometry and sharp constants.

Proved existence of extremal functions that saturate the inequality.

Analyzed the variational problem associated with Hardy-Morrey inequalities.

Abstract

Morrey's classical inequality implies the H\"older continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$ \lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le \int_\Omega |Du|^p \,dx $$ for any open set $\Omega\subsetneq \mathbb{R}^n$. This inequality is valid for functions supported in $\Omega$ and with $\lambda$ a positive constant independent of $u$. The crucial hypothesis is that the exponent $p$ exceeds the dimension $n$. This paper aims to develop a basic theory for this inequality and the associated variational problem. In particular, we study the relationship between the geometry of $\Omega$, sharp constants, and the existence of a nontrivial $u$ which saturates the inequality.