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

Abstract

We consider functions $u\in W_0^{m,1}(\Omega )$, 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 $\frac{{\partial }^{j}u(x)}{d(x{)}^{m-j-k}}\in W_0^{k,1}(\Omega )$ with

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 $l$.