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Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes

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Abstract

Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \({\mathbb{R}^N}\) , the class \({\mathcal{E}_{P,\{\omega\}}(\Omega)}\) of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case \({\mathcal{E}_{P,(\omega)}(\Omega)}\).

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References

  1. Bolley P., Camus J., Rodino L.: Analytic Gevrey hypoellipticity and iterates of operators. Rend. Sem. Mat. Univ. Politec. Torino. 45(3), 1–61 (1987)

    MATH  MathSciNet  Google Scholar 

  2. Bonet J., Meise R., Melikhov S.N.: A comparison of two different ways of define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)

    MATH  MathSciNet  Google Scholar 

  3. Bouzar C., Chaili L.: A Gevrey microlocal analysis of multi-anisotropic differential operators. Rend. Semin. Mat. Univ. Politec. Torino 64(3), 305–317 (2006)

    MATH  MathSciNet  Google Scholar 

  4. Braun R.W., Meise R., Taylor B.A.: Ultradifferentiable functions and Fourier analysis. Result. Math. 17, 206–237 (1990)

    MATH  MathSciNet  Google Scholar 

  5. Fernández C., Galbis A., Jornet D.: Pseudodifferential operators on non-quasianalytic classes of Beurling type. Studia Math. 167(2), 99–131 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Folland G.B.: Introducction to Partial Differential Operators. Princeton University Press, Princeton (1995)

    Google Scholar 

  7. Hörmander L.: On interior regularity of the solutions of partial differential equations. Comm. Pure Appl. Math. 11, 197–218 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  8. Hörmander L.: The Analysis of Linear Partial Differential Operators II. Springer-Verlag, Berlin (1990)

    MATH  Google Scholar 

  9. Komatsu H.: A characterization of real analytic functions. Proc. Jpn. Acad. 36, 90–93 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  10. Komatsu H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Tokyo Sec. IA 20, 25–105 (1973)

    MATH  MathSciNet  Google Scholar 

  11. Kotake T., Narasimhan M.S.: Regularitty theorems for fractional powers of a linear elliptic operator. Bull. Soc. Math. France 90, 449–471 (1962)

    MATH  MathSciNet  Google Scholar 

  12. Langenbruch M.: P-Funktionale und Randwerte zu hypoelliptischen Differentialoperatoren. Math. Ann. 239(1), 55–74 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  13. Langenbruch M.: Fortsetzung von Randwerten zu hypoelliptischen Differentialoperatoren und partielle Differentialgleichungen. J. Reine Angew. Math. 311/312, 57–79 (1979)

    MathSciNet  Google Scholar 

  14. Langenbruch M.: On the functional dimension of solution spaces of hypoelliptic partial differential operators. Math. Ann. 272, 217–229 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  15. Langenbruch M.: Bases in solution sheaves of systems of partial differential equations. J. Reine Angew. Math. 373, 1–36 (1987)

    MATH  MathSciNet  Google Scholar 

  16. Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)

    MATH  Google Scholar 

  17. Métivier G.: Propiété des itérés et ellipticité. Comm. Partial Differ. Equ. 9(3), 827–876 (1978)

    Article  Google Scholar 

  18. Mujica, J.: A completeness criterion for inductive limits of Banach spaces. In: Zapata, G.I. (ed.) Functional Analysis, Holomorphy and Approximation Theory II. North-Holland Mathematics Studies (1984)

  19. Newberger E., Zielezny Z.: The growth of hypoelliptic polynomials and Gevrey clases. Proc. Am. Math. Soc. 39(3), 547–552 (1973)

    MATH  MathSciNet  Google Scholar 

  20. Pérez-Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces. Noth-Holland Mathematics Studies, vol. 131 (1987)

  21. Wengenroth, J.: Derived functors in funcional analysis. In: Lecture Notes in Mathematics, vol. 1810. Springer-Verlag, Berlin (2003)

  22. Zanghirati L.: Iterates of a class of hypoelliptic operators and generalized Gevrey clases. Suppl B.U.M.I 1, 177–195 (1980)

    MATH  MathSciNet  Google Scholar 

  23. Zanghirati L.: Iterates of quasielliptic operators and Gevrey classes. Boll. U.M.I. 18B, 411–428 (1981)

    MathSciNet  Google Scholar 

  24. Zanghirati, L.: Iterati di operatori e regolarità Gevrey microlocale anisotropa. Rend. Sem. Mat. Univ. Padova 67 (1982)

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  1. Instituto de Matemática Pura y Aplicada IUMPA, Universidad Politécnica de Valencia, 46071, Valencia, Spain

    Jordi Juan-Huguet

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  1. Jordi Juan-Huguet
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Correspondence to Jordi Juan-Huguet.

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Jordi Juan-Huguet was partially supported by MEC and FEDER Project MTM2007-62643, Conselleria d’Educació de la GVA, Project GV/2010/040, grant F.P.U. AP-2006-04678 and the research net MTM2007-30904-E.

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Juan-Huguet, J. Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes. Integr. Equ. Oper. Theory 68, 263–286 (2010). https://doi.org/10.1007/s00020-010-1816-5

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  • DOI: https://doi.org/10.1007/s00020-010-1816-5

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