A Hardy type inequality for \(W^{m,1}_0(\Omega)\) functions (Q1932206)

The following result is proved: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary \(\partial\Omega\), and let \(d:\Omega \rightarrow(0,\infty)\) be a smooth function such that \(d(x)\) coincides with the distance of \(x\) to \(\partial\Omega\) when \(x\) is near \(\partial\Omega.\) Suppose there are given nonnegative integers \(m\), \(j\) and \(k\) such that \(m\geq2,\) \(1\leq k\leq m-1\) and \(1\leq j+k\leq m\). Then, denoting by \(\partial^{l}\) any partial derivative of order \(l,\) for every \(u\in W_{0}^{m,1}(\Omega)\) one has \(\partial^{j}u(x)/d(x)^{m-j-k}\in W_{0} ^{k,1}(\Omega)\) and \(\left\| \partial^{k}\left( \frac{\partial^{j} u(x)}{d(x)^{m-j-k}}\right) \right\| _{L^{1}(\Omega)}\leq C\left\| u\right\| _{W^{m,1}(\Omega)};\) here \(C\) is a positive constant that depends only on \(\Omega\) and \(m.\)