Semialgebraic Calderon-Zygmund theorem on regularization of the distance function

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