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

Final version