Pointwise Hardy inequalities

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.