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Extension operators for smooth functions on compact subsets of the reals. (English) Zbl 07238528

Summary: We introduce sufficient as well as necessary conditions for a compact set \(K\) such that there is a continuous linear extension operator from the space of restrictions \(C^\infty (K)=\{F|_K: F\in C^\infty (\mathbb{R})\}\) to \(C^\infty (\mathbb{R})\). This allows us to deal with examples of the form \(K=\{a_n:n\in \mathbb{N}\}\cup \{0\}\) for \(a_n\rightarrow 0\) previously considered by Fefferman and Ricci as well as Vogt.

MSC:

47A57 Linear operator methods in interpolation, moment and extension problems
46E25 Rings and algebras of continuous, differentiable or analytic functions
46A63 Topological invariants ((DN), (\(\Omega\)), etc.) for locally convex spaces
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[1] Bos, Len P.; Milman, Pierre D., Sobolev-Gagliardo-Nirenberg and Markov type inequalities on subanalytic domains, Geom. Funct. Anal., 5, 6, 853-923 (1995) · Zbl 0848.46022 · doi:10.1007/BF01902214
[2] Bierstone, Edward; Milman, Pierre D., Geometric and differential properties of subanalytic sets, Ann. Math. (2), 147, 3, 731-785 (1998) · Zbl 0912.32006 · doi:10.2307/120964
[3] Bierstone, Edward; Milman, Pierre D.; Pawłucki, Wiesław, Composite differentiable functions, Duke Math. J., 83, 3, 607-620 (1996) · Zbl 0868.32011 · doi:10.1215/S0012-7094-96-08318-0
[4] DeVore, Ronald A.; Lorentz, George G., Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1993), Berlin: Springer, Berlin · Zbl 0797.41016
[5] Fefferman, Charles, \(C^m\) extension by linear operators, Ann. Math. (2), 166, 3, 779-835 (2007) · Zbl 1161.46013 · doi:10.4007/annals.2007.166.779
[6] Frerick, Leonhard; Jordá, Enrique; Wengenroth, Jochen, Tame linear extension operators for smooth Whitney functions, J. Funct. Anal., 261, 3, 591-603 (2011) · Zbl 1232.46023 · doi:10.1016/j.jfa.2011.04.008
[7] Frerick, Leonhard; Jordá, Enrique; Wengenroth, Jochen, Whitney extension operators without loss of derivatives, Rev. Mat. Iberoam., 32, 2, 377-390 (2016) · Zbl 1348.47011 · doi:10.4171/RMI/888
[8] Fefferman, Charles; Ricci, Fulvio, Some examples of \(C^\infty\) extension by linear operators, Rev. Mat. Iberoam., 28, 1, 297-304 (2012) · Zbl 1242.47014 · doi:10.4171/RMI/678
[9] Frerick, Leonhard, Extension operators for spaces of infinite differentiable Whitney jets, J. Reine Angew. Math., 602, 123-154 (2007) · Zbl 1124.46014
[10] Goncharov, Alexander, A compact set without Markov’s property but with an extension operator for \(C^\infty \)-functions, Studia Math., 119, 1, 27-35 (1996) · Zbl 0857.46013 · doi:10.4064/sm-119-1-27-35
[11] Hörmander, Lars: The analysis of linear partial differential operators. I, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer, Berlin, Distribution theory and Fourier analysis (1990) · Zbl 0712.35001
[12] Malgrange, Bernard: Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, (1967) · Zbl 0177.17902
[13] Merrien, Jean, Prolongateurs de fonctions différentiables d’une variable réelle, J. Math. Pures Appl. (9), 45, 291-309 (1966) · Zbl 0163.06602
[14] Mitjagin, BS, Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk, 16, 4-100, 63-132 (1961) · Zbl 0104.08601
[15] Meise, Reinhold; Vogt, Dietmar, Introduction to functional analysis, Oxford Graduate Texts in Mathematics (1997), New York: The Clarendon Press, Oxford University Press, New York · Zbl 0924.46002
[16] Pawłucki, Wiesław, On the algebra of functions \(\mathscr{C}^k\)-extendable for each \(k\) finite, Proc. Am. Math. Soc., 133, 2, 481-484 (2005) · Zbl 1062.26025 · doi:10.1090/S0002-9939-04-07756-1
[17] Pawłucki, Wiesław; Pleśniak, Wiesław, Extension of \(C^\infty\) functions from sets with polynomial cusps, Studia Math., 88, 3, 279-287 (1988) · Zbl 0778.26010 · doi:10.4064/sm-88-3-279-287
[18] Stein, Elias M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series (1970), Princeton: Princeton University Press, Princeton · Zbl 0207.13501
[19] Tidten, Michael, Fortsetzungen von \(C^{\infty }\)-Funktionen, welche auf einer abgeschlossenen Menge in \({ R}^n\) definiert sind, Manuscripta Math., 27, 3, 291-312 (1979) · Zbl 0412.46027 · doi:10.1007/BF01309013
[20] Vogt, Dietmar, Restriction spaces of \(A^\infty \), Rev. Mat. Iberoam., 30, 1, 65-78 (2014) · Zbl 1327.46007 · doi:10.4171/RMI/769
[21] Vogt, Dietmar; Wagner, Max Josef, Charakterisierung der Quotientenräume von \(s\) und eine Vermutung von Martineau, Studia Math., 67, 3, 225-240 (1980) · Zbl 0464.46010 · doi:10.4064/sm-67-3-225-240
[22] Wengenroth, Jochen, Derived functors in functional analysis. Lecture Notes in Mathematics (2003), Berlin: Springer, Berlin · Zbl 1031.46001
[23] Whitney, Hassler, Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc., 36, 1, 63-89 (1934) · Zbl 0008.24902 · doi:10.1090/S0002-9947-1934-1501735-3
[24] Whitney, Hassler, On ideals of differentiable functions, Am. J. Math., 70, 635-658 (1948) · Zbl 0037.35502 · doi:10.2307/2372203
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