MATHEMATICA BOHEMICA, Vol. 124, No. 2–3, pp. 113-121 (1999)

On a higher-order Hardy inequality

David E. Edmunds, Jiri Rakosnik

D. E. Edmunds, Centre for Mathematical Analysis and Its Applications, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BN1 9QH, United Kingdom, e-mail:;
J. Rakosnik, Mathematical Institute, Academy of Sciences of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail:

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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