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Markov’s inequality and $$C^{\infty}$$ functions on sets with polynomial cusps. (English) Zbl 0579.32020
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 $$C^{\infty}({\mathbb{R}}^ n)$$.

##### MSC:
 32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables 32B20 Semi-analytic sets, subanalytic sets, and generalizations 26D05 Inequalities for trigonometric functions and polynomials 26D10 Inequalities involving derivatives and differential and integral operators 26E10 $$C^\infty$$-functions, quasi-analytic functions
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