# A Hardy type inequality for ${W}_{0}^{m,1}\left(\Omega \right)$ functions

Hernán Castro; Juan Dávila; Hui Wang

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 1, page 145-155
- ISSN: 1435-9855

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topCastro, Hernán, Dávila, Juan, and Wang, Hui. "A Hardy type inequality for $W^{m,1}_0(\Omega )$ functions." Journal of the European Mathematical Society 015.1 (2013): 145-155. <http://eudml.org/doc/277758>.

@article{Castro2013,

abstract = {We consider functions $u\in W^\{m,1\}_0(\Omega )$, where $\Omega \subset \mathbb \{R\}^N$ is a smooth bounded domain, and $m\ge 2$ is an integer. For all $j\ge 0, 1\le k\le m-1$, such that $1\le j+k\le m$, we prove that $\frac\{\partial ^iu(x)\}\{d(x)^\{m-j-k\}\}\in W^\{k,1\}_0(\Omega )$ with $\left\Vert \partial ^k(\frac\{\partial ^iu(x)\}\{d(x)^\{m-j-k\}\})_\{L^1(\Omega )\} \le C\right\Vert u \left\Vert _\{W^\{m,1\}(\Omega )\}\right.$, where $d$ is a smooth positive function which coincides with dist$(x,\partial \Omega )$ near $\partial \Omega $, and $\partial ^l$ denotes any partial differential operator of order $l$.},

author = {Castro, Hernán, Dávila, Juan, Wang, Hui},

journal = {Journal of the European Mathematical Society},

keywords = {Hardy inequality; Sobolev spaces; Hardy's inequality; Sobolev spaces},

language = {eng},

number = {1},

pages = {145-155},

publisher = {European Mathematical Society Publishing House},

title = {A Hardy type inequality for $W^\{m,1\}_0(\Omega )$ functions},

url = {http://eudml.org/doc/277758},

volume = {015},

year = {2013},

}

TY - JOUR

AU - Castro, Hernán

AU - Dávila, Juan

AU - Wang, Hui

TI - A Hardy type inequality for $W^{m,1}_0(\Omega )$ functions

JO - Journal of the European Mathematical Society

PY - 2013

PB - European Mathematical Society Publishing House

VL - 015

IS - 1

SP - 145

EP - 155

AB - We consider functions $u\in W^{m,1}_0(\Omega )$, where $\Omega \subset \mathbb {R}^N$ is a smooth bounded domain, and $m\ge 2$ is an integer. For all $j\ge 0, 1\le k\le m-1$, such that $1\le j+k\le m$, we prove that $\frac{\partial ^iu(x)}{d(x)^{m-j-k}}\in W^{k,1}_0(\Omega )$ with $\left\Vert \partial ^k(\frac{\partial ^iu(x)}{d(x)^{m-j-k}})_{L^1(\Omega )} \le C\right\Vert u \left\Vert _{W^{m,1}(\Omega )}\right.$, where $d$ is a smooth positive function which coincides with dist$(x,\partial \Omega )$ near $\partial \Omega $, and $\partial ^l$ denotes any partial differential operator of order $l$.

LA - eng

KW - Hardy inequality; Sobolev spaces; Hardy's inequality; Sobolev spaces

UR - http://eudml.org/doc/277758

ER -

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