Abstract: The Hardy inequality $\int_\Omega|u(x)|^pd(x)^{-p}\dd x\le c\int_\Omega|\nabla u(x)|^p\dd x$ with $d(x)=\operatorname{dist}(x,\partial\Omega)$ holds for $u\in C^\infty_0(\Omega)$ if $\Omega\subset\Bbb R^n$ is an open set with a sufficiently smooth boundary and if $1<p<\infty$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.
Keywords: Hardy inequality, capacity, $p$-thick set, maximal function, Sobolev space
Classification (MSC2000): 31C15, 46E35, 42B25
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