On a higher-order Hardy inequality

David Eric Edmunds; Jiří Rákosník

Mathematica Bohemica (1999)

  • Volume: 124, Issue: 2-3, page 113-121
  • ISSN: 0862-7959

Abstract

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The Hardy inequality ∫ Ω | u ( x ) | p d ( x ) - p x ¨ ≤ c ∫ Ω | ∇ u ( x ) | p x ¨ with d ( x ) = dist ( x , ∂ Ω ) holds for u ∈ C 0 ∞ ( Ω ) if Ω ⊂ ℝ n is an open set with a sufficiently smooth boundary and if 1 < p < ∞ . 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 .

How to cite

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Edmunds, David Eric, and Rákosník, Jiří. "On a higher-order Hardy inequality." Mathematica Bohemica 124.2-3 (1999): 113-121. <http://eudml.org/doc/248453>.

@article{Edmunds1999,
abstract = {The Hardy inequality $\int _\Omega |u(x)|^pd(x)^\{-p\}\ddot\{x\}\le c\int _\Omega |\nabla u(x)|^p\ddot\{x\}$ with $d(x)=\operatorname\{dist\}(x,\partial \Omega )$ holds for $u\in C^\infty _0(\Omega )$ if $\Omega \subset \mathbb \{R\}^n$ 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 $p=1$.},
author = {Edmunds, David Eric, Rákosník, Jiří},
journal = {Mathematica Bohemica},
keywords = {Hardy inequality; capacity; maximal function; Sobolev space; $p$-thick set; Hardy inequality; capacity; -thick set; maximal function; Sobolev space},
language = {eng},
number = {2-3},
pages = {113-121},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a higher-order Hardy inequality},
url = {http://eudml.org/doc/248453},
volume = {124},
year = {1999},
}

TY - JOUR
AU - Edmunds, David Eric
AU - Rákosník, Jiří
TI - On a higher-order Hardy inequality
JO - Mathematica Bohemica
PY - 1999
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 124
IS - 2-3
SP - 113
EP - 121
AB - The Hardy inequality $\int _\Omega |u(x)|^pd(x)^{-p}\ddot{x}\le c\int _\Omega |\nabla u(x)|^p\ddot{x}$ with $d(x)=\operatorname{dist}(x,\partial \Omega )$ holds for $u\in C^\infty _0(\Omega )$ if $\Omega \subset \mathbb {R}^n$ 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 $p=1$.
LA - eng
KW - Hardy inequality; capacity; maximal function; Sobolev space; $p$-thick set; Hardy inequality; capacity; -thick set; maximal function; Sobolev space
UR - http://eudml.org/doc/248453
ER -

References

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