A conjecture of Palamodov about the functors Extk in the category of locally convex spaces

AbstractIn 1971 Palamodov proved that in the category of locally convex spaces the derived functors Extk(E,X) of Hom(E,·) all vanish if E is a (DF)-space, X is a Fréchet space, and one of them is nuclear. He conjectured a “dual result”, namely that Extk(E,X)=0 for all k∈N if E is a metrizable locally convex space, X is a complete (DF)-space, and one of them is nuclear. Assuming the continuum hypothesis we give a complete answer to this conjecture: If X is an infinite-dimensional nuclear (DF)-space, then (1)There is a normed space E such that Ext1(E,X)≠0.(2)Ext2(KN,X)≠0 where KN is a countable product of lines.(3)Extk(E,X)=0 for all k⩾3 and all locally convex spaces E

Extension operators for smooth functions on compact subsets of the reals

[EN] 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-infinity (K) = {F vertical bar(K) : F is an element of C-infinity (R)} to C-infinity (R). This allows us to deal with examples of the form K = {a(n) : n is an element of N} boolean OR{0} for a(n) -> 0 previously considered by Fefferman and Ricci as well as Vogt.The research of all authors was partially supported by GVA AICO/2016/054 . The research of the second author was partially supported by the project MTM2016-76647-P.Frerick, L.; Jorda Mora, E.; Wengenroth, J. (2020). Extension operators for smooth functions on compact subsets of the reals. Mathematische Zeitschrift. 295(3-4):1537-1552. https://doi.org/10.1007/s00209-019-02388-5S153715522953-4Bos, Len P., Milman, Pierre D.: Sobolev–Gagliardo–Nirenberg and Markov type inequalities on subanalytic domains. Geom. Funct. Anal. 5(6), 853–923 (1995)Bierstone, Edward, Milman, Pierre D.: Geometric and differential properties of subanalytic sets. Ann. Math. (2) 147(3), 731–785 (1998)Bierstone, Edward, Milman, Pierre D., Pawłucki, Wiesław: Composite differentiable functions. Duke Math. J. 83(3), 607–620 (1996)DeVore, Ronald A., Lorentz, George G.: Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)Fefferman, Charles: $$C^m$$ extension by linear operators. Ann. Math. (2) 166(3), 779–835 (2007)Frerick, Leonhard, Jordá, Enrique, Wengenroth, Jochen: Tame linear extension operators for smooth Whitney functions. J. Funct. Anal. 261(3), 591–603 (2011)Frerick, Leonhard, Jordá, Enrique, Wengenroth, Jochen: Whitney extension operators without loss of derivatives. Rev. Mat. Iberoam. 32(2), 377–390 (2016)Fefferman, Charles, Ricci, Fulvio: Some examples of $$C^\infty$$ extension by linear operators. Rev. Mat. Iberoam. 28(1), 297–304 (2012)Frerick, Leonhard: Extension operators for spaces of infinite differentiable Whitney jets. J. Reine Angew. Math. 602, 123–154 (2007)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)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)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)Merrien, Jean: Prolongateurs de fonctions différentiables d’une variable réelle. J. Math. Pures Appl. (9) 45, 291–309 (1966)Mitjagin, B.S.: Approximate dimension and bases in nuclear spaces. Uspehi Mat. Nauk 16(4 (100)), 63–132 (1961)Meise, Reinhold, Vogt, Dietmar: Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press, Oxford University Press, New York (1997). Translated from the German by M. S. Ramanujan and revised by the authorsPawłucki, Wiesław: On the algebra of functions $$\mathscr {C}^k$$-extendable for each $$k$$ finite. Proc. Am. Math. Soc. 133(2), 481–484 (2005). (Electronic)Pawłucki, Wiesław, Pleśniak, Wiesław: Extension of $$C^\infty$$ functions from sets with polynomial cusps. Studia Math. 88(3), 279–287 (1988)Stein, Elias M.: Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)Tidten, Michael: Fortsetzungen von $$C^{\infty }$$-Funktionen, welche auf einer abgeschlossenen Menge in $${ R}^{n}$$ definiert sind. Manuscripta Math. 27(3), 291–312 (1979)Vogt, Dietmar: Restriction spaces of $$A^\infty$$. Rev. Mat. Iberoam. 30(1), 65–78 (2014)Vogt, Dietmar, Wagner, Max Josef: Charakterisierung der Quotientenräume von $$s$$ und eine Vermutung von Martineau. Studia Math. 67(3), 225–240 (1980)Wengenroth, Jochen: Derived functors in functional analysis. Lecture Notes in Mathematics, vol. 1810. Springer, Berlin (2003)Whitney, Hassler: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934)Whitney, Hassler: On ideals of differentiable functions. Am. J. Math. 70, 635–658 (1948

Strongly continuous semigroups on some Fréchet spaces

We prove that for a strongly continuous semigroup T on the Frechet space omega of all scalar sequences, its generator is a continuous linear operator A is an element of L(omega) and that, for all x is an element of omega and t >= 0, the series exp(tA)(x) = Sigma(infinity)(k=0) t(k)/k! A(k)(x) converges to T-t(x). This solves a problem posed by Conejero. Moreover, we improve recent results of Albanese, Bonet, and Ricker about semigroups on strict projective limits of Banach spaces. (C) 2013 Elsevier Inc. All rights reserved.The research was partially done during a stay of the fourth named author at EPSA-UPV. This visit was supported by Proyecto Prometeo 11/2013/013. The research of the first and second named authors was supported by MICINN and FEDER, Project MTM2010-15200. The research of the second named author was partially supported by Programa de Apoyo a la Investigacion y Desarrollo de la UPV PAID-06-12.Frerick, L.; Jorda Mora, E.; Kalmes, T.; Wengenroth, J. (2014). Strongly continuous semigroups on some Fréchet spaces. Journal of Mathematical Analysis and Applications. 412(1):121-124. https://doi.org/10.1016/j.jmaa.2013.10.053S121124412

Construction and Test of a Cryocatcher Prototype for SIS100

The main accelerator SIS100 of the FAIR-complex will provide heavy ion beams of highest intensities. Beam loss due to ionization is the most demanding loss mechanism at operation with high intensity, intermediate charge state heavy ions. A special synchrotron design has been devel- oped for SIS100, aiming for hundred percent control of ion- ization beam loss by means of a dedicated cryogenic ion catcher system. To suppress dynamic vacuum effects, the cryo catcher system shall provide a significantly reduced effective desorption yield. The construction and test of a prototype cryocatcher is a task of the EU-FP-7 workpack- age COLMAT. A prototype test setup, including cryostat has been constructed, manufactured and tested under real- istic conditions with beams from the heavy ion synchrotron SIS18. The design and results are presented