A Hardy type inequality for \(W^{m,1}_0(\Omega)\) functions
type: Item
instance of: scholarly article
title: A Hardy type inequality for \(W^{m,1}_0(\Omega)\) functions
zbMATH Open document ID: 1273.26030
DOI: 10.4171/JEMS/357
author: Hernán Castro
author: Juan Dávila
author: Hui Wang
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2013-01-17T00:00:00Z
review text: The following result is proved: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary \(\partial\Omega\), and let \(d:\Omega \rightarrow(0,\infty)\) be a smooth function such that \(d(x)\) coincides with the distance of \(x\) to \(\partial\Omega\) when \(x\) is near \(\partial\Omega.\) Suppose there are given nonnegative integers \(m\), \(j\) and \(k\) such that \(m\geq2,\) \(1\leq k\leq m-1\) and \(1\leq j+k\leq m\). Then, denoting by \(\partial^{l}\) any partial derivative of order \(l,\) for every \(u\in W_{0}^{m,1}(\Omega)\) one has \(\partial^{j}u(x)/d(x)^{m-j-k}\in W_{0} ^{k,1}(\Omega)\) and \(\left\| \partial^{k}\left( \frac{\partial^{j} u(x)}{d(x)^{m-j-k}}\right) \right\| _{L^{1}(\Omega)}\leq C\left\| u\right\| _{W^{m,1}(\Omega)};\) here \(C\) is a positive constant that depends only on \(\Omega\) and \(m.\)
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6126558
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: Sobolev spaces
MaRDI profile type: Publication
cites work: Q5318420
cites work: Functional analysis, Sobolev spaces and partial differential equations
cites work: Q4538008
cites work: A Hardy type inequality for \(W^{m,1}(0,1)\) functions
cites work: Elliptic partial differential equations of second order
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full work available at URL: https://doi.org/10.4171/jems/357
OpenAlex ID: W4246943608
description: scientific article; zbMATH DE number 6126558