Semialgebraic Calderón-Zygmund theorem on regularization of the distance function

We prove that, for any closed semialgebraic subset W of R n and for any positive integer p , there exists a Nash function f : R n \ W ⟶ ( 0 , ∞ ) which is equivalent to the distance function from W and at the same time it is Λ p -regular in the sense that | D α f ( x ) | ≤ C d ( x , W ) 1 - | α | , for each x ∈ R n \ W and each α ∈ N n such that 1 ≤ | α | ≤ p , where C is a positive constant. In particular, f is Lipschitz. Some applications of this result are given.