Markov's inequality and \(C^{\infty}\) functions on sets with polynomial cusps
We say that a subset E of \({\mathbb{R}}^ n\) is uniformly polynomially cuspidal (UPC) if there exist positive constants M and m, and a positive integer d such that for each point \(x\in \bar E\), one may choose a polynomial map \(h_ x: {\mathbb{R}}\to {\mathbb{R}}^ n\) of degree at most d satisfying the following conditions: \(h_ x((0,1])\subset E\) and \(h_ x(0)=x\), \(dist(h_ x(t),{\mathbb{R}}^ n\setminus E)\geq Mt^ m\) for all x in \(\bar E\) and \(t\in [0,1]\). Every bounded convex domain in \({\mathbb{R}}^ n\) and every bounded domain with Lipschitz boundary are UPC. Using Hironaka's rectilinearization theorem and \({\L}ojasiewicz's\) inequality we show that every bounded subanalytic set in \({\mathbb{R}}^ n\) such that \(E\subset \overline{int E}\) is UPC. We also give examples of UPC sets that are not subanalytic. There are two main purposes of this paper. The first one is to state Markov's inequality on a UPC subset E of \({\mathbb{R}}^ n:\) There exists a constant \(r>0\) such that for every polynomial \(p: {\mathbb{R}}^ n\to {\mathbb{R}}\) of degree at most k and for every multiindex \(\alpha \in {\mathbb{Z}}^ n_+\), \(\sup_{E}| D^{\alpha}p| \leq Ck^{r| \alpha |}\sup_{E}| p|,\) where C is a constant depending only on E and \(\alpha\). The second purpose is to prove the following version of Bernstein's theorem: If E is a UPC compact set in \({\mathbb{R}}^ n\), then a function \(f: E\to {\mathbb{R}}\) extends to a \(C^{\infty}\) function \(\tilde f\) on \({\mathbb{R}}^ n\) iff for each \(r>0\), \(\lim_{k\to \infty} k^ rdist_ E(f,P_ k)=0,\) where \(P_ k\) is the linear space of (the restrictions to E of) all polynomials from \({\mathbb{R}}^ n\) to \({\mathbb{R}}\) of degree at most k, and \(dist_ E(f,P_ k):=\inf \{\sup_{E}| f-p|:\quad p\in P_ k\}.\) Our main tool is a Hölder continuity property of Siciak's extremal function of \(E: \Phi_ E(x)=\sup_{k\geq 1} \sup \{| p(x)|^{1/k}: p\in P_ k, \sup_{E\quad}| p| \leq 1\}.\) Both result will be used in our forthcoming paper to give a simple construction of a continuous linear extension operator from the space of \(C^{\infty}\) functions on a UPC subset of \({\mathbb{R}}^ n\) to the space \(C^{\infty}({\mathbb{R}}^ n)\).
bounded subanalytic set in \({\mathbb{R}}^ n\)
scientific article; zbMATH DE number 3927349
scientific article