On a Hardy–Morrey inequality

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