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

Abstract

We consider functions $u\in W^{m,1}_0(\Omega)$, where $\Omega\subset \mathbb R^N$ is a smooth bounded domain, and $m\geq 2$ is an integer. For all $j\geq 0$, $1\leq k\leq m-1$, such that $1\leq j+k\leq m$, we prove that $\frac{\partial^ju(x)}{d(x)^{m-j-k}}\in W^{k,1}_0(\Omega)$ 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 $l$.