On a Hardy-Morrey inequality

Abstract

Morrey's classical inequality implies the Hölder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality $$\hat{I}»\Vert \frac{u}{d_{\hat{I}}{\text{c◯}}^{1-n/p}}{\Vert }_{\infty }^p\le {\int }_{\hat{I}}\text{c◯}|Du{|}^{p}dx$$ for any open set $\hat{I}\text{c◯}⊊{\mathbb{R}}^n$. This inequality is valid for functions supported in $\hat{I}\text{c◯}$ and with $\hat{I}»$ 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 $\hat{I}\text{c◯}$, sharp constants, and the existence of a nontrivial $u$ which saturates the inequality.