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1997 | 126 | 2 | 171-198
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On the range of convolution operators on non-quasianalytic ultradifferentiable functions

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Let $ℇ_{(ω)}(Ω)$ denote the non-quasianalytic class of Beurling type on an open set Ω in $ℝ^n$. For $μ ∈ ℇ'_{(ω)}(ℝ^n)$ the surjectivity of the convolution operator $T_μ: ℇ_{(ω)}(Ω_1) → ℇ_{(ω)}(Ω_2)$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $(Ω_1, Ω_2)$ and the existence of a fundamental solution for μ or equivalently by a slowly decreasing condition for the Fourier-Laplace transform of μ. Similar conditions characterize the surjectivity of a convolution operator $S_μ: D'_{{ω}}(Ω_1) → D'_{{ω}}(Ω_2)$ between ultradistributions of Roumieu type whenever $μ ∈ ℇ'_{{ω}}(ℝ^n)$. These results extend classical work of Hörmander on convolution operators between spaces of $C^∞$-functions and more recent one of Ciorănescu and Braun, Meise and Vogt.
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  • Dpto. Análisis Matemático, Universidad de Valencia, E-46100 Burjasot (Valencia), Spain, galbis@uv.es
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Bibliografia
  • [1] C. A. Berenstein and M. A. Dostal, Analytically Uniform Spaces and Their Applications to Convolution Equations, Lecture Notes in Math. 256, Springer, 1972.
  • [2] K. D. Bierstedt, R. Meise and B. H. Summers, A projective description of weighted inductive limits, Trans. Amer. Math. Soc. 272 (1982), 107-160.
  • [3] J. Bonet and A. Galbis, The range of non-surjective convolution operators on Beurling spaces, Glasgow Math. J. 38 (1996), 125-135.
  • [4] J. Bonet, A. Galbis and S. Momm, Nonradial Hörmander algebras of several variables, manuscript.
  • [5] R. W. Braun, An extension of Komatsu's second structure theorem for ultradistributions, J. Fac. Sci. Univ. Tokyo 40 (1993), 411-417.
  • [6] W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), 206-237.
  • [7] W. Braun, R. Meise and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultradifferentiable functions, Proc. London Math. Soc. 61 (1990), 344-370.
  • [8] Chou, La Transformation de Fourier Complexe et l'Équation de Convolution, Lecture Notes in Math. 325, Springer, 1973.
  • [9] I. Ciorănescu, Convolution equations in ω-ultradistribution spaces, Rev. Roumaine Math. Pures Appl. 25 (1980), 719-737.
  • [10] L. Ehrenpreis, Solution of some problems of division, Part IV. Invertible and elliptic operators, Amer. J. Math. 82 (1960), 522-588.
  • [11] U. Franken and R. Meise, Generalized Fourier expansions for zero-solutions of surjective convolution operators on D'ℝ and $D'_ω ℝ$, Note Mat. 10, Suppl. 1 (1990), 251-272.
  • [12] O. v. Grudzinski, Konstruktion von Fundamentallösungen für Convolutoren, Manuscripta Math. 19 (1976), 283-317.
  • [13] S. Hansen, Das Fundamentalprinzip für Systeme linearer partieller Differentialgleichungen mit konstanten Koeffizienten, Habilitationsschrift, Paderborn, 1982.
  • [14] L. Hörmander, On the range of convolution operators, Ann. of Math. 76 (1962), 148-170.
  • [15] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Princeton Univ. Press, 1967.
  • [16] L. Hörmander, The Analysis of Linear Partial Differential Operators I, II, Springer, 1983.
  • [17] H. Komatsu, Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Univ. Tokyo 20 (1973), 25-105.
  • [18] M. Langenbruch, Surjective partial differential operators on spaces of ultradifferentiable functions of Roumieu type, Results Math. 29 (1996), 254-275.
  • [19] R. Meise and B. A. Taylor, Whitney's extension theorem for ultradifferentiable functions of Beurling type, Ark. Mat. 26 (1988), 265-287.
  • [20] R. Meise, B. A. Taylor and D. Vogt, Equivalence of slowly decreasing conditions and local Fourier expansions, Indiana Univ. Math. J. 36 (1987), 729-756.
  • [21] R. Meise, B. A. Taylor and D. Vogt, Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, Math. Nachr. 180 (1996), 213-242.
  • [22] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Univ. Press, 1997.
  • [23] T. Meyer, Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type, Studia Math. 125 (1997), 101-129.
  • [24] S. Momm, Closed ideals in nonradial Hörmander algebras, Arch. Math. (Basel) 58 (1992), 47-55.
  • [25] S. Momm, Division problems in spaces of entire functions of finite order, in: Functional Analysis, K. D. Bierstedt, A. Pietsch, W. Ruess and D. Vogt (eds.), Marcel Dekker, 1993, 435-457.
  • [26] S. Momm, A Phragmén-Lindelöf theorem for plurisubharmonic functions on cones in $ℂ^N$, Indiana Univ. Math. J. 41 (1992), 861-867.
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