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

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

Annales de l'institut Fourier (1990)

  • Volume: 40, Issue: 3, page 619-655
  • ISSN: 0373-0956

Abstract

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

How to cite

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Taylor, B. A., Meise, R., and Vogt, Dietmar. "Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse." Annales de l'institut Fourier 40.3 (1990): 619-655. <http://eudml.org/doc/74890>.

@article{Taylor1990,
abstract = {Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P(D)$ are characterized that admit a continuous linear right inverse on $\{\cal E\}(\Omega )$ or $\{\cal D\}^\{\prime \}(\Omega )$, $\Omega $ an open set in $\{\bf R\}^ n$. For bounded $\Omega $ with $C^ 1$-boundary these properties are equivalent to $P(D)$ being very hyperbolic. For $\Omega =\{\bf R\}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.},
author = {Taylor, B. A., Meise, R., Vogt, Dietmar},
journal = {Annales de l'institut Fourier},
keywords = {constant coefficient partial differential operators; continuous linear right inverse; Phragmén-Lindelöf condition},
language = {eng},
number = {3},
pages = {619-655},
publisher = {Association des Annales de l'Institut Fourier},
title = {Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse},
url = {http://eudml.org/doc/74890},
volume = {40},
year = {1990},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 3
SP - 619
EP - 655
AB - Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P(D)$ are characterized that admit a continuous linear right inverse on ${\cal E}(\Omega )$ or ${\cal D}^{\prime }(\Omega )$, $\Omega $ an open set in ${\bf R}^ n$. For bounded $\Omega $ with $C^ 1$-boundary these properties are equivalent to $P(D)$ being very hyperbolic. For $\Omega ={\bf R}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.
LA - eng
KW - constant coefficient partial differential operators; continuous linear right inverse; Phragmén-Lindelöf condition
UR - http://eudml.org/doc/74890
ER -

References

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  1. [A] L.V. AHLFORS, Conformal Invariants ; Topics in Geometric Function Theory, McGraw - Hill, 1973. Zbl0272.30012
  2. [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. Zbl0726.46022
  3. [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. Zbl0848.35023
  4. [C1] D.K. COHOON, Nonexistence of a continuous right inverse for parabolic differential operators, J. Diff. Equ., (1969), 503 - 511. Zbl0183.38701MR40 #3364
  5. [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. Zbl0235.35018MR47 #5659
  6. [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. 
  7. [H] S. HANSEN, On the fundamental principle of L. Ehrenpreis, Banach Center Publ., 10 (1983), 185 - 201. Zbl0555.35009MR85h:35054
  8. [HO1] L. HÖRMANDER, Linear Partial Differential Operators, Springer, 1963. Zbl0108.09301
  9. [HO2] L. HÖRMANDER, On the existence of real analytic solutions of partial differential equations with constant coefficients, Invent. Math., 21 (1973), 151 - 182. Zbl0282.35015MR49 #817
  10. [HO3] L. HÖRMANDER, The Analysis of Linear Partial Differential Operators, Vol. II, Springer, 1983. Zbl0521.35001
  11. [M] E. MICHAEL, Continuous selections I, Ann. Math., 63 (1956), 361 - 382. Zbl0071.15902MR17,990e
  12. [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. Zbl0649.46031MR89m:35042
  13. [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. Zbl0709.46018MR92b:46122
  14. [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. Zbl0745.32004
  15. [MTV4] R. MEISE, B.A. TAYLOR and D. VOGT, Phragmén-Lindelöf principles for algebraic varieties, preprint. Zbl0896.32008
  16. [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. Zbl0858.46030
  17. [MTV6] R. MEISE, B.A. TAYLOR and D. VOGT, Seminar report, November 1989. 
  18. [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. Zbl0607.42011MR88k:47042
  19. [P] V.P. PALAMODOV, A criterion for splitness of differential complexes with constant coefficients, preprint. Zbl1112.58304
  20. [S] L. SCHWARTZ, Théorie des distributions, Hermann, 1966. Zbl0149.09501
  21. [T1] F. TREVES, Locally Convex Spaces and Linear Partial Differential Equations, Springer, 1967. Zbl0152.32104MR36 #6986
  22. [T2] F. TREVES, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach, 1966. Zbl0164.40602
  23. [V1] D. VOGT, On the solvabilty of P(D)f = g for vector valued functions, RIMS Kokyuroku, 508 (1983), 168 - 182. 
  24. [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). Zbl0585.46060MR86i:46075
  25. [VA] B.L. VAN der WAERDEN, Algebra I, Springer Grundlehren, 33-6 (1964). Zbl0067.00501MR31 #1292

Citations in EuDML Documents

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  1. Rüdiger Braun, A partial differential operator which is surjective on Gevrey classes Γ d ( ℝ ³ ) with 1 ≤ d < 2 and d ≥ 6 but not for 2 ≤ d < 6
  2. Rüdiger W. Braun, The surjectivity of a constant coefficient homogeneous differential operator in the real analytic functions and the geometry of its symbol
  3. Rüdiger W. Braun, Reinhold Meise, B. A. Taylor, A new characterization of the analytic surfaces in ℂ 3 that satisfy the local Phragmén-Lindelöf condition
  4. Rüdiger Braun, Reinhold Meise, B. Taylor, A radial Phragmén-Lindelöf estimate for plurisubharmonic functions on algebraic varieties
  5. Uwe Franken, Reinhold Meise, Extension and lacunas of solutions of linear partial differential equations

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