Semialgebraic Calderon-Zygmund theorem on regularization of the distance function

Abstract

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