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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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[1] Baouendi, M.S., Goulaouic, C.: Approximation polynomiale de fonctionsC et analytiques. Ann. Inst. Fourier, Grenoble21, 149–173 (1971) · Zbl 0215.17503
[2] Baouendi, M.S., Goulaouic, C.: Approximation of analytic functions on compact sets and Bernstein’s inequality. Trans. Am. Math. Soc.189, 251–261 (1974) · Zbl 0296.41016
[3] Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math.149, 1–40 (1982) · Zbl 0547.32012
[4] Bernstein, S.N.: Collected works. I. Moskva: Akad. Nauk. SSSR 1952 (Russian)
[5] Bierstone, E.: Differentiable functions. Bol. Soc. Bras. Mat.11, 139–190 (1980) · Zbl 0584.58006
[6] Denkowska, Z., Łojasiewicz, S., Stasica, J.: Certaines propriétés élémentaires des ensembles sous-analytiques. Bull. Acad. Pol. Sci. Sér. Sci. Math.27, 529–536 (1979) · Zbl 0435.32006
[7] Denkowska, Z., Łojasiewicz, S., Stasica, J.: Sur le théorème du complémentaire pour les ensembles sous-analytiques. Bull. Acad. Pol. Sci. Sér. Sci. Math.27, 537–539 (1979) · Zbl 0457.32003
[8] Gabrielov, A.M.: Projections of semianalytic sets. Funkt. Anal. Priloz.2, 18–30 (1968) (Russian) · Zbl 0179.08503
[9] Goetgheluck, P.: Inégalité de Markov dans les ensembles effilés. J. Approx. Theory30, 149–154 (1980) · Zbl 0457.41015
[10] Hironaka, H.: Subanalytic sets. In: Number theory, algebraic geometry and commutative algebra (in honor of Y. Akizuki) pp. 453–493. Tokyo: Kinokuniya 1973
[11] Hironaka, H.: Introduction to real-analytic sets and real-analytic maps. Pisa: Istituto Matematico ”L. Tonelli” 1973 · Zbl 0297.32008
[12] Klimek, M.: Extremal plurisubharmonic functions and L-regular sets in \(\mathbb{C}\) n . Proc. R. Ir. Acad. Sect. A82, 217–230 (1982) · Zbl 0494.32005
[13] Leja, F.: Sur les suites de polynômes, les ensembles fermés et la fonction de Green. Ann. Soc. Pol. Math.12, 57–71 (1934) · Zbl 0010.20103
[14] Łojasiewicz, S.: Ensembles semi-analytiques. Bures-sur-Yvette: Inst. Hautes Etudes Sci. 1964
[15] Mazurkiewicz, S.: Les fonctions quasi-analytiques dans l’espace fonctionnel. Mathematica (Cluj)13, 16–21 (1937) · Zbl 0018.13403
[16] Pawłucki, W., Pleśniak, W.: Markov’s inequality on subanalytic sets. Proc. Alfred Haar Memorial Conf. Budapest 1985. In: Colloq. Math. Soc. Janos Bolyai 49 (to appear) · Zbl 0617.41011
[17] Pleśniak, W.: Quasianalytic functions in the sence of Bernstein. Dissertationes Math. (Rozprawy Mat.)147, Warszawa: PWN 1977
[18] Pleśniak, W.: Invariance of the L-regularity under holomorphic mappings. Trans. Am. Math. Soc.246, 373–383 (1978)
[19] Pleśniak, W.: Sur la L-régularité des compacts de \(\mathbb{C}\) n . Toulouse: Séminaire d’analyse complexe de Toulouse, Université Paul Sabatier, U.E.R. Math. Inf. Gestion 1980
[20] Pleśniak, W.: L-regularity of subanalytic sets in \(\mathbb{R}\) n . Bull. Pol. Acad. Sci. Math.32, 647–651 (1984) · Zbl 0563.32007
[21] Pleśniak, W.: Again on Markov’s inequality. In: Constructive Theory of Functions’84, 679–683. Sofia: Bulgarian Acad. Sci 1984 · Zbl 0591.26007
[22] Rahman, Q.I., Schmeisser, G.: Les inégalités de Markoff et de Bernstein. Montréal: Les Presses de l’Université de Montréal 1983
[23] Rudin, W.: Function theory in the unit ball of \(\mathbb{C}\) n . Berlin, Heidelberg, New York: Springer 1980 · Zbl 0495.32001
[24] Siciak, J.: On some extremal functions and their applications in the theory of analytic functions of several complex variables. Trans. Am. Math. Soc.105, 322–357 (1962) · Zbl 0111.08102
[25] Siciak, J.: Extremal plurisubharmonic functions in \(\mathbb{C}\) n . Ann. Pol. Math.39, 175–211 (1981) · Zbl 0477.32018
[26] Siciak, J.: Highly noncontinuable functions on polynomially convex sets. Univ. Iagello. Acta Math.25, 95–107 (1985) · Zbl 0585.32012
[27] Siciak, J.: Highly noncontinuable functions on polynomially convex sets. In: Lect. Not. Math. 1094. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0551.32013
[28] Stasica, J.: Whitney’s condition for subanalytic sets. Zesz. Nauk. Uniw. Jagiellon, Pr. Mat.23, 211–221 (1982) · Zbl 0508.32001
[29] Timan, A.F.: Theory of approximation of functions of a real variable. Oxford: Pergamon Press 1963 · Zbl 0117.29001
[30] Tougeron, J.C.: Idéaux de fonctions différentiables. Berlin, Heidelberg, New York: Springer 1972
[31] Zakharyuta, V.P.: Extremal plurisubharmonic functions, orthogonal polynomials, and the Bernstein-Walsh theorem for analytic functions of several complex variables. Ann. Pol. Math.33, 137–148 (1976) (Russian) · Zbl 0341.32011
[32] Zerner, M.: Développement en série de polynômes orthonormaux des fonctions indéfiniment différentiables. C.R. Acad. Sci. Paris268, 218–220 (1969) · Zbl 0189.14601
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