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Continuous linear right inverses for differential operators

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  • C. v. Ossietzky-Universität Oldenburg, FB6-Mathematik, Postfach 2503, D-29111 Oldenburg, Germany

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Bibliografia

[BT] C. A. Berenstein and B. A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math. 33 (1979), 109-143.
[BS] E. Bierstone and G. W. Schwartz, Continuous linear division and extension of $C^∞$ functions, Duke Math. J. 50 (1983), 233-271.
[BBMT] J. Bonet, R. W. Braun, R. Meise and B. A. Taylor, Whitney's extension theorem for nonquasianalytic classes of ultradifferentiable functions, Studia Math. 99 (1991), 155-184.
[BMT] J. Bonet, R. Meise and B. A. Taylor, Whitney's extension theorem for ultradifferentiable functions of Roumieu type, Proc. Roy. Irish. Acad. 89A (1989), 53-66.
[BMT] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), 207-237.
[BMV] R. W. Braun, B. 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.
[B] J. Bruna, An extension theorem of Whitney type for non-quasianalytic classes of functions, J. London Math. Soc. (2) 22 (1980), 495-505.
[C1] D. K. Cohoon, Nonexistence of a continuous linear right inverse for parabolic differential operators, J. Differential Equations 6 (1969), 503-511.
[C2] D. K. Cohoon, Nonexistence of a continuous linear right inverse for linear partial differential operators with constant coefficients, Math. Scand. 29 (1971), 337-342.
[E] L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Van Nostrand, 1967.
[F] U. Franken, Stetige lineare Ausdehnungsoperatoren auf Räumen ultradifferenzierbarer Funktionen vom Beurling Typ, Dissertation, Düsseldorf, 1992.
[H] F. Haslinger, Weighted spaces of entire functions, Indiana Univ. Math. J. 35 (1986), 193-208.
[Hö1] L. Hörmander, An Introduction to Complex Analysis in Several Variables, North-Holland, 1973.
[Hö2] L. Hörmander, The Analysis of Linear Partial Differential Operators I + II, Grundlehren Math. Wiss. 256+257, Springer, Berlin, 1983.
[Hö3] L. Hörmander, Generators of some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943-949.
[KM] Yu. F. Korobeĭnik and S. N. Melikhov, Linear continuous right inverse for representation operator and application to convolution operator, preprint.
[K] H. Komatsu, Ultradistributions I. Structure theorem and a characterization, J. Fac. Sci. Tokyo Ser. IA 20 (1973), 25-105.
[L1] M. Langenbruch, Partial differential equations without solution operators in weighted spaces of (generalized) functions, Manuscripta Math. 56 (1986), 353-374.
[L2] M. Langenbruch, Solution operators for partial differential equations in weighted Gevrey spaces, Michigan Math. J. 35 (1990), 3-24.
[L3] M. Langenbruch, Complemented kernels for partial differential operators on weighted spaces of generalized functions, Studia Math. 89 (1988), 37-63.
[L4] M. Langenbruch, Real roots of polynomials and right inverses for partial differential operators in the space of tempered distributions, Proc. Roy. Soc. Edinburgh 114A (1990), 169-179.
[L5] M. Langenbruch, Tame right inverses for partial differential equations, in: Advances in the Theory of Fréchet Spaces, T. Terzioglu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, 1989, 79-114.
[L6] M. Langenbruch, Continuous linear right inverses for convolution operators in spaces of real analytic functions, Studia Math. 110 (1994), 65-82.
[L7] M. Langenbruch, Splitting of the ∂̅-complex in weighted spaces of square integrable functions, Rev. Mat. Univ. Complut. Madrid 5 (1992), 201-223.
[L8] M. Langenbruch, The splitting condition for the weighted ∂̅-complex, Result. Math. 22 (1992), 560-597.
[L9] M. Langenbruch, Differentiable functions and the ∂̅-complex, in: Functional Analysis: Proceedings of the Essen Conference, K. D. Bierstedt, A. Pietsch, W. M. Ruess and D. Vogt (eds.), Lecture Notes in Pure and Appl. Math. 150, Dekker, 1993, 415-434.
[L10] M. Langenbruch, Power series spaces and weighted solution spaces of partial differential equations, Math. Z. 194 (1987), 71-88.
[L11] M. Langenbruch, Ultradifferentiable functions on compact intervals, Math. Nachr. 37 (1989), 21-45.
[L12] M. Langenbruch, Bases in spaces of ultradifferentiable functions with compact support, Math. Ann. 281 (1988), 31-42.
[L13] M. Langenbruch, Extension of ultradifferentiable functions, Manuscripta Math. 83 (1994), 123-143.
[L14] M. Langenbruch and S. Momm, Complemented submodules in weighted spaces of analytic functions, Math. Nachr. 157 (1992), 263-276.
[M1] R. Meise, Structure of closed linear translation invariant subspaces of A(ℂ) and kernels of analytic convolution operators, in: Functional Analysis: Surveys and Recent Results III, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland, 1984, 331-347.
[M2] R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals, J. Reine Angew. Math. 363 (1985), 59-95.
[MMT] R. Meise, S. Momm and B. A. Taylor, Splitting of slowly decreasing ideals in weighted algebras of entire functions, in: Complex Analysis II, C. A. Berenstein (ed.), Lecture Notes in Math. 1276, Springer, Berlin, 1987, 229-252.
[MT1] R. Meise and B. A. Taylor, Each nonzero convolution operator on the entire functions admits a continuous linear right inverse, Math. Z. 197 (1988), 139-152.
[MT2] R. Meise and B. A. Taylor, Splitting of closed ideals in (DFN)-algebras of entire functions and the property (DN), Trans. Amer. Math. Soc. 302 (1987), 341-370.
[MT3] R. Meise and B. A. Taylor, Sequence space representations for (FN)-algebras of entire functions modulo closed ideals, Studia Math. 85 (1987), 203-227.
[MT4] R. Meise and B. A. Taylor, A decomposition lemma for entire functions and its application to spaces of ultradifferentiable functions, Math. Nachr. 142 (1989), 45-72.
[MT5] R. Meise and B. A. Taylor, Opérateurs linéaires continus d'extension pour les fonctions ultradifférentiables sur des intervalles compacts, C. R. Acad. Sci. Paris Sér. I 302 (1986), 219-222.
[MT6] R. Meise and B. A. Taylor, Whitney's extension theorem for ultradifferentiable functions of Beurling type, Ark. Mat. 26 (1988), 265-287.
[MT7] R. Meise and B. A. Taylor, Linear extension operators for ultradifferentiable functions of Beurling type on compact sets, Amer. J. Math. 111 (1989), 309-337.
[MTV1] R. Meise, B. A. Taylor and D. Vogt, Caractérisation des opérateurs linéaires aux dérivées partielles avec coefficients constants sur $E(ℝ^n)$ admettant un inverse à droite qui est linéaire et continu, C. R. Acad. Sci. Paris Sér. I 307 (1988), 239-242.
[MTV2] R. Meise, B. A. Taylor and D. Vogt, Partial differential operators with continuous linear right inverse, in: Advances in the Theory of Fréchet spaces, T. Terzioglu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, 1989, 47-62.
[MTV3] R. Meise, B. A. Taylor and D. Vogt, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier (Grenoble) 40 (1990), 619-655.
[MTV4] R. Meise, B. A. Taylor and D. Vogt, Equivalence of analytic and plurisubharmonic Phragmen-Lindelöf principles on algebraic variaties, in: Proc. Sympos. Pure Math. 52, Amer. Math. Soc. 1991, 287-308.
[MTV5] R. Meise, B. A. Taylor and D. Vogt, Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, preprint.
[MTV6] R. Meise, B. A. Taylor and D. Vogt, Phragmen-Lindelöf principles for algebraic varieties, J. Amer. Math. Soc., to appear.
[MTV7] R. Meise, B. A. Taylor and D. Vogt, Extremal plurisubharmonic functions of linear growth on algebraic varieties, Math. Z., to appear.
[MTV8] R. Meise, B. A. Taylor and D. Vogt, Continuous linear right inverses for partial differential operators with constant coefficients and Phragmen-Lindelöf conditions, in: Functional Analysis: Proceedings of the Essen Conference, K. D. Bierstedt, A. Pietsch, W. M. Ruess and D. Vogt (eds.), Lecture Notes in Pure and Appl. Math. 150, Dekker, 1993, 357-389.
[MV1] R. Meise and D. Vogt, Characterization of convolution operators on spaces of $C^∞$-functions admitting a continuous linear right inverse, Math. Ann. 279 (1987), 141-155.
[MV2] R. Meise and D. Vogt, Einführung in die Funktionalanalysis, Braunschweig, Wiesbaden, 1992.
[Mi] E. Michael, Continuous selections I, Ann. of Math. 63 (1956), 361-382.
[Mo1] S. Momm, Ideale in gewichteten Algebren holomorpher Funktionen auf dem Einheitskreis, Dissertation, Düsseldorf, 1988.
[Mo2] S. Momm, Partial differential operators of infinite order with constant coefficients on the space of analytic functions on the polydisc, Studia Math. 96 (1990), 52-71.
[Mo3] S. Momm, Convex univalent functions and continuous linear right inverses, J. Funct. Anal. 103 (1992), 85-103.
[Mo4] S. Momm, Convolution equations on the analytic functions on convex domains in the plane, Bull. Sci. Math. 118 (1994), 259-270.
[Mo5] S. Momm, A critical growth rate for the pluricomplex Green function, Duke Math. J. 72 (1993), 487-502.
[Mo6] S. Momm, On the dependence of analytic solutions of partial differential equations on the right hand side, Trans. Amer. Math. Soc. 345 (1994), 729-752.
[Mo7] S. Momm, Linear topological invariants and splitting of closed ideals in weighted algebras of analytic functions on the disc, Results in Math. 17 (1990), 128-139.
[Mo8] S. Momm, Complemented ideals in weighted algebras of analytic functions on the disc, Michigan Math. J. 37 (1990), 259-274.
[Mo9] S. Momm, Closed principal ideals in nonradial Hörmander algebras, Arch. Math. (Basel) 58 (1992), 47-55.
[Pa1] V. P. Palamodov, On a Stein manifold the Dolbeault complex splits in positive dimensions, Math. USSR-Sb. 17 (1972), 289-316.
[Pa2] V. P. Palamodov, A criterion for splitness of differential complexes with constant coefficients, in: Geometrical and Algebraical Aspects in Several Complex Variables, C. A. Berenstein and D. C. Struppa (eds.), Editel, 1991, 265-291.
[Pe] H. J. Petzsche, On E. Borel's theorem, Math. Ann. 282 (1988), 299-313.
[PV] M. Poppenberg and D. Vogt, A tame splitting theorem for exact sequences of Fréchet spaces, Math. Z., to appear.
[S] K. Schwerdtfeger, Faltungsoperatoren auf Räumen holomorpher und beliebig oft differenzierbarer Funktionen, Dissertation, Düsseldorf, 1982.
[T] B. A. Taylor, Linear extension operators for entire functions, Michigan Math. J. 29 (1982), 185-197.
[Tr1] F. Trèves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966.
[Tr2] F. Trèves, Locally Convex Spaces and Linear Partial Differential Equations, Springer, 1967.
[V1] D. Vogt, On the solvability of P(D)f=g for vector valued functions, RIMS Kokyoroku 508 (1983), 168-182.
[V2] D. Vogt, Some results on continuous linear maps between Fréchet spaces, in: Functional Analysis: Surveys and Recent Results II, K. D. Bierstedt and B. Fuchssteiner (eds.), North-Holland Math. Stud. 90, North-Holland, 1984, 349-381.
[V3] D. Vogt, Subspaces and quotient spaces of (s), in: Functional Analysis: Surveys and Recent Results, K.-D. Bierstedt and B. Fuchssteiner (eds.), North-Holland Math. Stud. 27, North-Holland, 1977, 167-187.
[V4] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301.
[V5] D. Vogt, Sequence space representations of spaces of test functions and distributions, in: Functional Analysis, Holomorphy and Approximation Theory, G. I. Zapata (ed.), Lecture Notes in Pure and Appl. Math. 83 (1983), 405-443.
[MV] D. Vogt und M. J. Wagner, Characterisierung der Unterräume und Quotientenräume der nuklearen stabilen Potenzreihenräume von unendlichem Typ, Studia Math. 70 (1981), 63-80.

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