Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

Taylor, B. A.; Meise, R.; Vogt, Dietmar

Annales de l'Institut Fourier, Tome 40 (1990) no. 3, pp. 619-655.

Nous résolvons complètement un problème de L. Schwartz sur la caractérisation des opérateurs différentiels aux dérivées partielles $P\left(D\right)$, à coefficients constants sur un ouvert $\Omega$ de ${\mathbf{R}}^n$, qui admettent un inverse à droite linéaire continu sur $ℰ\left(\Omega \right)$ ou ${𝒟}^{\prime }\left(\Omega \right)$. Si $\Omega$ est borné à frontière de classe $C^1$, ces propriétés sont équivalentes à une hyperbolicité très forte de $P\left(D\right)$. Si $\Omega ={\mathbf{R}}^n$, elles sont équivalentes à la validité d’un principe du type de Phragmén-Lindelöf sur la variété des zéros du polynôme $P$.

Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P\left(D\right)$ are characterized that admit a continuous linear right inverse on $ℰ\left(\Omega \right)$ or ${𝒟}^{\prime }\left(\Omega \right)$, $\Omega$ an open set in ${\mathbf{R}}^n$. For bounded $\Omega$ with $C^1$-boundary these properties are equivalent to $P\left(D\right)$ being very hyperbolic. For $\Omega ={\mathbf{R}}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.

[A] L.V. Ahlfors, Conformal Invariants ; Topics in Geometric Function Theory, McGraw - Hill, 1973. | Zbl

[BMV1] R.W. Braun, R.Meise and D. Vogt, 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

[BMV2] R.W.Braun, R.Meise and D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type, manuscrit. | Zbl

[C1] D.K. Cohoon, Nonexistence of a continuous right inverse for parabolic differential operators, J. Diff. Equ., (1969), 503 - 511. | MR | Zbl

[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. | MR | Zbl

[CR] M.L. De Cristoforis, 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] S. Hansen, On the fundamental principle of L. Ehrenpreis, Banach Center Publ., 10 (1983), 185 - 201. | MR | Zbl

[HO1] L. Hörmander, Linear Partial Differential Operators, Springer, 1963. | Zbl

[HO2] L. Hörmander, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151 - 182. | MR | Zbl

[HO3] L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. II, Springer, 1983. | Zbl

[M] E. Michael, Continuous selections I, Ann. Math., 63 (1956), 361 - 382. | MR | Zbl

[MTV1] R. Meise, B.A. Taylor and D. Vogt, 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

[MTV2] R. Meise, B.A. Taylor and D. Vogt, 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

[MTV3] R. Meise, B.A. Taylor and D. Vogt, Equivalence of analytic and plurisubharmonic Phragmén-Lindelöf principles on algebraic varieties, to appear in Proceedings of Symposia in Pure Mathematics. | Zbl

[MTV4] R. Meise, B.A. Taylor and D. Vogt, Phragmén-Lindelöf principles for algebraic varieties, preprint. | Zbl

[MTV5] R. Meise, B.A. Taylor and D. Vogt, Continuous linear right inverses for partial differential operators on non-quasianalytic classes and on ultradistributions, manuscript. | Zbl

[MTV6] R. Meise, B.A. Taylor and D. Vogt, Seminar report, November 1989.

[MV] R. Meise, D. Vogt, Characterization of convolution operators on spaces of C∞-functions admitting a continuous linear right inverse, Math. Ann., 279 (1987), 141 - 155. | MR | Zbl

[P] V.P. Palamodov, A criterion for splitness of differential complexes with constant coefficients, preprint. | Zbl

[S] L. Schwartz, Théorie des distributions, Hermann, 1966.

[T1] F. Treves, Locally Convex Spaces and Linear Partial Differential Equations, Springer, 1967. | MR | Zbl

[T2] F. Treves, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966. | Zbl

[V1] D. Vogt, On the solvabilty of P(D)f = g for vector valued functions, RIMS Kokyuroku, 508 (1983), 168 - 182.

[V2] D. Vogt, 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] B.L. Van Der Waerden, Algebra I, Springer Grundlehren, 33-6 (1964). | MR | Zbl