Pointwise Hardy inequalities

Abstract

If $\Omega \subset {{\mathbb R}}^{n}$ is an open set with the sufficiently regular boundary, then the Hardy inequality $\int _{\Omega }|u|^{p}\varrho ^{-p}\leq C\int _{\Omega }|\nabla u|^{p}$ holds for $u\in C_{0}^{\infty }(\Omega )$ and $1<p<\infty$, where $\varrho (x)=\operatorname {dist}(x,\partial \Omega )$. The main result of the paper is a pointwise inequality $|u|\leq \varrho M_{2\varrho }|\nabla u|$, where on the right hand side there is a kind of maximal function. The pointwise inequality combined with the Hardy–Littlewood maximal theorem implies the Hardy inequality. This generalizes some recent results of Lewis and Wannebo.