Results in Mathematics

, Volume 67, Issue 3–4, pp 431–444 | Cite as

A Class of Invertible Subelliptic Operators in S(m, g)-Classes

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Abstract

Given a self-adjoint second order differential operator L with positive characteristic and subellipticity of order 1 ≤ τ < 2. In this paper we study the invertibility of L + C in a suitable S(m,g)-class.

Keywords

Degenerate elliptic operators nonhomogeneous calculus microlocal analysis 

Mathematics Subject Classification

Primary 35J70 Secondary 35A27 47G30 
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References

  1. 1.
    Bony, J.M., Chemin, J.Y.: Espaces fonctionnels associés au calcul de Weyl–Hörmander. Bull. Soc. Math. Fr. 122, 77–118 (1994)MATHMathSciNetGoogle Scholar
  2. 2.
    Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44(1), 45–57 (1977)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Beals, R., Fefferman, C.: Spatially inhomogeneous pseudodifferential operators I. Commun. Pure Appl. Math. 27, 1–24 (1974)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Beals, R., Gaveau, B., Greiner, P.: Green’s functions for some highly degenerate elliptic operators. J. Funct. Anal. 165, 407–429 (1999)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bony, J.-M., Lerner, B.: Quantification asymptotique et microlocalisation d’ordre superieur I. Ann. Sci. Ec. Norm. Sup. 22, 377–433 (1989)MATHMathSciNetGoogle Scholar
  6. 6.
    Cancelier, C.E., Chemin, J.-Y., Xu, C.-J.: Calcul de Weyl–Hörmander et opérateurs sous-elliptiques. Ann. Inst. Fourier 43, 1157–1178 (1993)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Delgado, J.: Estimations L p pour une classe d’opérateurs pseudo-différentiels dans le cadre du calcul de Weyl–Hörmander. J. Anal. Math. 100, 337–374 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Delgado, J., Zamudio, A.: Invertibility for a class of degenerate elliptic operators. J. Pseudo-Differ. Oper. Appl. 1, 207–231 (2010)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Fefferman, C., Phong, D.H.: On positivity of pseudo-differential operators. Proc. Natl. Acad. Sci. USA 75(10), 4673–4674 (1978)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. III. Springer, Berlin (1985)Google Scholar
  11. 11.
    Lerner, N.: Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators. Pseudo-Differential Operators. Birkhäuser, Basel (2010)CrossRefMATHGoogle Scholar
  12. 12.
    Maniccia, L., Mughetti, M.: Parametrix construction for a class of anisotropic operators. Ann. Univ. Ferrara. 49, 263–284 (2003)MATHMathSciNetGoogle Scholar
  13. 13.
    Maniccia, L., Mughetti, M.: SAK principle for a class of Grushin-type operators. Rev. Math. Iberoam. 22(1), 259–286 (2006)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Mughetti, M., Nicola, F.: On the generalization of Hörmander’s inequality. Commun. PDE 30(4–6), 509–537 (2005)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mustapha, S.: Sous-ellipticité dans le cadre du calcul S(m, g). Commun. PDE 19, 245–275 (1994)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Mustapha, S.: Sous-ellipticité dans le cadre du calcul S(m, g) II. Commun. PDE 20, 541–566 (1995)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Xu, C., Zhu, X.: On the inverse of a class of degenerate elliptic operator. Chinese J. Contemp. Math. 16(3), 261–274 (1995)MathSciNetGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

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