Solving a problem of L. Schwartz, those constant coefficient partial differential operators are characterized that admit a continuous linear right inverse on or , an open set in . For bounded with -boundary these properties are equivalent to being very hyperbolic. For they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial .
Nous résolvons complètement un problème de L. Schwartz sur la caractérisation des opérateurs différentiels aux dérivées partielles , à coefficients constants sur un ouvert de , qui admettent un inverse à droite linéaire continu sur ou . Si est borné à frontière de classe , ces propriétés sont équivalentes à une hyperbolicité très forte de . Si , elles sont équivalentes à la validité d’un principe du type de Phragmén-Lindelöf sur la variété des zéros du polynôme .
@article{AIF_1990__40_3_619_0, author = {Taylor, B. A. and Meise, R. and Vogt, Dietmar}, title = {Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse}, journal = {Annales de l'Institut Fourier}, pages = {619--655}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {40}, number = {3}, year = {1990}, doi = {10.5802/aif.1226}, zbl = {0703.46025}, mrnumber = {92e:46083}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1226/} }
TY - JOUR AU - Taylor, B. A. AU - Meise, R. AU - Vogt, Dietmar TI - Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse JO - Annales de l'Institut Fourier PY - 1990 SP - 619 EP - 655 VL - 40 IS - 3 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1226/ DO - 10.5802/aif.1226 LA - en ID - AIF_1990__40_3_619_0 ER -
%0 Journal Article %A Taylor, B. A. %A Meise, R. %A Vogt, Dietmar %T Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse %J Annales de l'Institut Fourier %D 1990 %P 619-655 %V 40 %N 3 %I Institut Fourier %C Grenoble %U https://aif.centre-mersenne.org/articles/10.5802/aif.1226/ %R 10.5802/aif.1226 %G en %F AIF_1990__40_3_619_0
Taylor, B. A.; Meise, R.; Vogt, Dietmar. Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse. Annales de l'Institut Fourier, Volume 40 (1990) no. 3, pp. 619-655. doi : 10.5802/aif.1226. https://aif.centre-mersenne.org/articles/10.5802/aif.1226/
[A] Conformal Invariants ; Topics in Geometric Function Theory, McGraw - Hill, 1973. | Zbl
,[BMV1] Applications of the projective limit functor to convolution equations and partial differential operators, pp. 29 - 46 in Advances in the Theory of Fréchet Spaces, T. Terzioğlu (Ed.), NATO ASI Series C Vol. 287 (Kluwer), 1989. | Zbl
, and ,[BMV2] Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type, manuscrit. | Zbl
, and ,[C1] Nonexistence of a continuous right inverse for parabolic differential operators, J. Diff. Equ., (1969), 503 - 511. | MR | Zbl
,[C2] Nonexistence of a continuous linear right inverse for linear partial differential operators with constant coefficients, Math. Scand., 29 (1971), 337-342. | MR | Zbl
,[CR] Soluzioni con lacune di certi operatori differenziali lineari, Rendiconti Accademia Nazionale delle Scienze detta dei XL, Memorie di Matematica, 102, vol. VIII (1984), 137 - 142.
,[H] On the fundamental principle of L. Ehrenpreis, Banach Center Publ., 10 (1983), 185 - 201. | MR | Zbl
,[HO1] Linear Partial Differential Operators, Springer, 1963. | Zbl
,[HO2] On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151 - 182. | MR | Zbl
,[HO3] The Analysis of Linear Partial Differential Operators, Vol. II, Springer, 1983. | Zbl
,[M] Continuous selections I, Ann. Math., 63 (1956), 361 - 382. | MR | Zbl
,[MTV1] Caractérisation des opérateurs linéaires aux dérivées partielles avec coefficients constants sur ε(ℝn) admettant un inverse à droite qui est linéaire et continu, C. R. Acad. Paris, 307 (1988), 239 - 242. | MR | Zbl
, and ,[MTV2] Partial differential operators with continuous linear right inverse, pp. 47 - 62 in Advances in the Theory of Fréchet Spaces, T. Terzioğlu (Ed.), NATO ASI Series C Vol. 287 (Kluwer), 1989. | MR | Zbl
, and ,[MTV3] Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf principles on algebraic varieties, to appear in Proceedings of Symposia in Pure Mathematics. | Zbl
, and ,[MTV4] Phragmén-Lindelöf principles for algebraic varieties, preprint. | Zbl
, and ,[MTV5] Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, manuscript. | Zbl
, and ,[MTV6] Seminar report, November 1989.
, and ,[MV] Characterization of convolution operators on spaces of C∞-functions admitting a continuous linear right inverse, Math. Ann., 279 (1987), 141 - 155. | MR | Zbl
, ,[P] A criterion for splitness of differential complexes with constant coefficients, preprint. | Zbl
,[S] Théorie des distributions, Hermann, 1966.
,[T1] Locally Convex Spaces and Linear Partial Differential Equations, Springer, 1967. | MR | Zbl
,[T2] Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966. | Zbl
,[V1] On the solvabilty of P(D)f = g for vector valued functions, RIMS Kokyuroku, 508 (1983), 168 - 182.
,[V2] Some results on continuous linear maps between Fréchet spaces, p. 349 - 381 in Functional Analysis : Surveys and Recent Results II ; K.D. Bierstedt and B. Fuchssteiner (Eds), North-Holland Mathematics Studies, 90 (1984). | MR | Zbl
,[VA] Algebra I, Springer Grundlehren, 33-6 (1964). | MR | Zbl
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