Semialgebraic Calderón-Zygmund theorem on regularization of the distance function

We prove that, for any closed semialgebraic subset W of \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ and for any positive integer p, there exists a Nash function \(f:{\mathbb {R}}^n\setminus W\longrightarrow (0, \infty )\) $f:{\mathbb{R}}^n\backslash W⟶(0,\infty )$ which is equivalent to the distance function from W and at the same time it is \(\Lambda _p\) ${\mathrm{\Lambda }}_p$ -regular in the sense that \(|D^\alpha f(x)|\le C d(x, W)^{1- |\alpha |}\) $|D^{\alpha }{f(x)|\le Cd(x,W)}^{1-|\alpha |}$ , for each \(x\in {\mathbb {R}}^n{\setminus } W\) $x\in {\mathbb{R}}^n\backslash W$ and each \(\alpha \in {\mathbb {N}}^n\) $\alpha \in {\mathbb{N}}^n$ such that \(1\le |\alpha |\le p\) $1\le |\alpha |\le p$ , where C is a positive constant. In particular, f is Lipschitz. Some applications of this result are given.