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Closed ideals in algebras of smooth functions

Seria
Rozprawy Matematyczne tom/nr w serii: 371 wydano: 1997
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Warianty tytułu
Abstrakty
EN
Abstract
A topological algebra admits spectral synthesis of ideals (SSI) if every closed ideal in this algebra is an intersection of closed primary ideals. According to classical results this is the case for algebras of continuous, several times continuously differentiable, and Lipschitz functions. New examples (and counterexamples) of function algebras that admit or fail to have SSI are presented. It is shown that the Sobolev algebra $W_p^{l}(ℝⁿ)$, 1 ≤ p < ∞, has the property of SSI for and only for n = 1 and 2 ≤ n < p. It is also proved that every algebra $C^m Lip φ$ in one variable admits SSI. A unified approach to SSI based on an abstract spectral synthesis theorem for a class of Banach algebras (called D-algebras) defined in terms of point derivations and consisting of functions with "order of smoothness" not greater than 1 is discussed. Within this framework, theorems on SSI for Zygmund algebras $Λ_φ$ in one and two variables not imbedded in C¹ as well as for their separable counterparts $λ_φ$ are obtained. The fact that a Zygmund algebra $Λ_φ$ is a D-algebra is equivalent to a special extension theorem of independent interest which leads to a solution of the spectral approximation problem for the algebras $Λ_φ$ in the cases mentioned above. Closed primary ideals and point derivations in arbitrary Zygmund algebras are described.
EN
CONTENTS
Introduction........................................................................................5
1. Main definitions and basic examples..............................................7
2. Closed ideals in Sobolev algebras...............................................10
 2.0. Notation...................................................................................10
 2.1. Preliminary observations and results.......................................11
 2.2. Closed primary ideals..............................................................13
 2.3. Spectral synthesis of ideals.....................................................15
3. Spectral synthesis of ideals in the algebras $C^m Lip φ$............18
4. D-algebras...................................................................................21
5. Zygmund algebras.......................................................................26
 5.1. Basic properties.......................................................................26
 5.2. Extensions, approximations, and traces...................................32
 5.3. Closed primary ideals...............................................................40
 5.4. Point derivations......................................................................43
 5.5. An extension property and spectral synthesis..........................46
 5.6. Proof of Theorem 5.1...............................................................48
Appendix..........................................................................................52
 1. Traces of generalized Lipschitz spaces.......................................53
 2. Traces of Zygmund spaces.........................................................58
 3. Proof of Proposition 5.2.11..........................................................62
References.......................................................................................65
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 371
Liczba stron
67
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCLXXI
Daty
wydano
1997
otrzymano
1996-10-09
poprawiono
1997-02-20
Twórcy
  • Department of Mathematics, Idaho State University, Pocatello, Idaho 83209, U.S.A. , hanin@isu.edu
Bibliografia
  • [1] R. A. Adams, Sobolev Spaces, Academic Press, 1975.
  • [2] J. J. Benedetto, Spectral Synthesis, Academic Press, New York, 1975.
  • [3] A. Beurling, On the spectral synthesis of bounded functions, Acta Math. 81 (1949), 225-238.
  • [4] A. Beurling, Sur les spectres des fonctions, in: Analyse Harmonique, Colloq. Internat. CNRS 15, Paris, 1949, 9-29.
  • [5] Yu. A. Brudnyĭ, A multidimensional analog of a theorem of Whitney, Math. USSR-Sb. 11 (1970), 157-170.
  • [6] Yu. A. Brudnyĭ and P. A. Shvartsman, A description of the trace of a function from the generalized Lipschitz space on an arbitrary compact set, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl' State Univ., Yaroslavl', 1982, 16-24 (in Russian).
  • [7] J. Bruna, Spectral synthesis in non-quasi-analytic classes of infinitely differentiable functions, Bull. Sci. Math. (2) 104 (1980), 65-78.
  • [8] V. I. Burenkov, On the approximations of functions in Sobolev spaces by finite functions on an arbitrary open set, Soviet Math. Dokl. 13 (1972), 60-63.
  • [9] E. M. Dyn'kin, An asymptotic Cauchy problem for the Laplace equation, Ark. Mat. 34 (1996), 245-264.
  • [10] E. M. Dyn'kin, An asymptotic Cauchy problem for the Laplace equation. Hölder-Zygmund scale, to appear.
  • [11] E. M. Dyn'kin and L. G. Hanin, Spectral synthesis of ideals in Zygmund algebras: the asymptotic Cauchy problem approach, Michigan Math. J. 43 (1996), 539-557.
  • [12] V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials, Nauka, Moscow, 1977 (in Russian).
  • [13] R. E. Edwards, Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York, 1965.
  • [14] E. Gagliardo, Proprietèa di alcune classi di funzioni in pièu variabili, Ricerche Mat. 7 (1958), 102-137.
  • [15] I. M. Gel'fand, D. A. Raikov and G. E. Shilov, Commutative Normed Rings, Chelsea, Bronx, N.Y., 1964.
  • [16] G. Glaeser, Étude de quelques algèebres tayloriennes, J. Analyse Math. 6 (1958), 1-124.
  • [17] G. Glaeser, Synthèese spectrale des idéaux de fonctions lipschitziennes, C. R. Acad. Sci. Paris 260 (1965), 1539-1542.
  • [18] L. G. Hanin, Spectral synthesis of ideals in algebras of functions having generalized derivatives, Russian Math. Surveys 39 (2) (1984), 167-168.
  • [19] L. G. Hanin, Closed primary ideals and point derivations in Zygmund algebras, in: Constructive Theory of Functions, Proceedings, Publishing House of the Bulgarian Academy of Sciences, Sofia, 1984, 397-402.
  • [20] L. G. Hanin, A theorem on spectral synthesis of ideals for a class of Banach algebras, Soviet Math. Dokl. 35 (1) (1987), 108-112.
  • [21] L. G. Hanin, Description of traces of functions with higher order derivatives satisfying a generalized Zygmund condition on an arbitrary closed set, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl' State Univ., Yaroslavl', 1988, 128-144 (in Russian).
  • [22] L. G. Hanin, The structure of closed ideals in some algebras of smooth functions, Amer. Math. Soc. Transl. 149 (1991), 97-113.
  • [23] L. I. Hedberg, Spectral synthesis in Sobolev spaces, and uniqueness of solutions of the Dirichlet problem, Acta Math. 147 (1981), 237-264.
  • [24] L. I. Hedberg and T. H. Wolff, Thin sets in nonlinear potential theory, Ann. Inst. Fourier (Grenoble) 33 (4) (1983), 161-187.
  • [25] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer, Berlin, 1963.
  • [26] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 2, Springer, Berlin, 1970.
  • [27] A. Jonsson and H. Wallin, Function Spaces on Subsets of ℝⁿ, Math. Rep. 2 (1) 1984.
  • [28] J.-P. Kahane, Séries de Fourier absolument convergentes, Springer, Berlin, 1970.
  • [29] E. Lasker, Zur Theorie der Moduln und Ideale, Math. Ann. 60 (1905), 20-116.
  • [30] J. Lindenstrauss, On nonlinear projections in Banach spaces, Michigan Math. J. 11 (1964), 263-287.
  • [31] B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966.
  • [32] P. Malliavin, Sur l'impossibilité de la synthèese spectrale dans une algèebre de fonctions presque périodiques, C. R. Acad. Sci. Paris 248 (1959), 1756-1759.
  • [33] P. Malliavin, Sur l'impossibilité de la synthèese spectrale sur la droite, C. R. Acad. Sci. Paris 248 (1959), 2155-2157.
  • [34] P. Malliavin, Impossibilité de la synthèese spectrale sur les groupes abéliens non compacts, Hautes Études Sci. Publ. Math. 2 (1959), 85-92.
  • [35] A. Marchaud, Sur les dérivées et sur les différences des fonctions de variables réelles, J. Math. Pures Appl. 6 (1927), 337-425.
  • [36] V. G. Maz'ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Uspekhi Mat. Nauk 38 (3) (1983), 23-86 (in Russian).
  • [37] E. J. McShane, Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
  • [38] E. Noether, Idealtheorie in Ringbereichen, Math. Ann. 83 (1921), 24-66.
  • [39] N. M. Osadchiĭ, Algebras L²ₙ(Γ) and the structure of closed ideals in these algebras, Ukrain. Mat. Zh. 26 (5) (1974), 669-670 (in Russian).
  • [40] L. Schwartz, Théorie générale des fonctions moyenne-périodiques, Ann. of Math. 48 (1947), 857-929.
  • [41] L. Schwartz, Sur une propriété de synthèese spectrale dans les groupes non compacts, C. R. Acad. Sci. Paris 227 (1948), 424-426.
  • [42] D. R. Sherbert, The structure of ideals and point derivations in Banach algebras of Lipschitz functions, Trans. Amer. Math. Soc. 111 (1964), 240-272.
  • [43] G. E. Shilov, On regular normed rings, Trudy Mat. Inst. Steklov. 21 (1947), 1-118 (in Russian).
  • [44] E. E. Shnol', Closed ideals in the ring of continuously differentiable functions, Mat. Sb. 27 (1950), 281-284 (in Russian).
  • [45] P. A. Shvartsman, On the trace of functions of two variables that satisfy the Zygmund condition, in: Studies in the Theory of Functions of Several Real Variables, Yaroslavl' State Univ., Yaroslavl', 1982, 145-168 (in Russian).
  • [46] P. A. Shvartsman, Traces of functions of Zygmund class, Siberian Math. J. 28 (1987), 848-865.
  • [47] S. L. Sobolev, On a theorem in functional analysis, Mat. Sb. 4 (3) (1938), 471-497 (in Russian).
  • [48] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
  • [49] M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375-481.
  • [50] R. S. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060.
  • [51] L. Waelbroeck, Closed ideals of Lipschitz functions, in: Proc. Internat. Sympos. ``Function Algebras", F. Birtel (ed.), Scott-Foresmann, Chicago, 1966, 322-325.
  • [52] H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63-89.
  • [53] H. Whitney, On ideals of differentiable functions, Amer. J. Math. 70 (1948), 635-658.
  • [54] A. Zygmund, Smooth functions, Duke Math. J. 12 (1945), 47-76.
Języki publikacji
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Uwagi
1991 Mathematics Subject Classification: Primary 46E25, 46E35, 46H10, 46J10, 46J15, 46J20; Secondary 26A16, 41A10.
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0012-3862
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