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

We consider functions $u\in {W}_{0}^{m,1}\left(\Omega \right)$, 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 $\frac{{\partial}^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\in {W}_{0}^{k,1}\left(\Omega \right)$ with $\u2225{\partial}^{k}{\left(\frac{{\partial}^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\right)}_{{L}^{1}\left(\Omega \right)}\le C\u2225u\u2225{}_{{W}^{m,1}\left(\Omega \right)}$, 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 $l$.