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Pointwise Hardy inequalities. (English) Zbl 0911.31005

A compact subset \(E\subset \mathbb{R}^n\) is called uniformly \(p\)-thick, \(1<p<\infty\), if there is a \(b>0\) such that the capacity condition \[ C_{1,p} (\overline{B}(x,r)\cap E,B(x,2r))\geq bC_{1,p} (\overline{B}(x,r), B(x,2r)) \] holds for any \(x\in E\) and \(0<r<\infty\). Let \(\Omega\subset \mathbb{R}^n\) be an open and proper subset whose complement is uniformly \(p\)-thick. It is proved that then there is a \(q\), \(1<q<p\), such that \[ | u(x)| \rho(x)^{-1}\leq CM_{2\rho(x),q} |\nabla u|(x) \] for \(u\in C_0^\infty(\Omega)\) where \[ M_{R,q}g(x)= \sup_{r\leq R}\Biggl( \frac{1}{| B(x,r)|} \int_{B(x,r)}| g(z)|^q dz\Biggr). \] Combined with the Hardy-Littlewood maximal theorem, this pointwise inequality implies the Hardy inequality. As noted in the paper, this generalizes some results by Lewis and Wannebo.

MSC:

31C15 Potentials and capacities on other spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
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