arXiv subjects
Publications and source records attributed to Wiesław Pawłucki.
We give a proof of the o-minimal version of the Whitney Extension Theorem simplified as compared to the original ones. A new simplifying ingredient is a definable variant of Urysohn's lemma for class $\mathcal{C}^q$ (see Section 3).
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 $Λ_p$-regular in the sense that $|D^αf(x)|\leq C d(x, W)^{1- |α|}$, for each $x\in \mathbb{R}^n\setminus W$ and each $α\in \mathbb{N}^n$ such that $1\leq |α|\leq p$, where $C$ is a positive constant. In particular, $f$ is Lipschitz. Some applications of this result are given.