A Hardy type inequality for $W_0^{m,1}\left(\Omega \right)$ functions

Castro, 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 -