An application of symbol calculus

  • Emmanuel Andronikof


Let X = ℂ t,x 1+n , t ∈ ℂ, x = (x 1,…,x n ) ∈ ℂ n , and let (t, x; τ,ξ)) be the associated symplectic coordinates in T*X. In Kashiwara and Oshima’s study of regular systems (cf [5]), the following definition occurs (with a slightly different vocabulary): a matrix of microdifferential operators A(x,D x ) is essentially of order ≥ 0 if there exists ν > 0 such that the coefficients of any power of A are microdifferential operators of order at most v. It is shown in [5] that any regular system of microdifferential equations with regular singularities along V = t1 = … = ξτ = 0, τ ≠ 0, is a quotient of a system of the form (tD t A(x, D x ))u = D x1 u = … = D u = 0, with A essentially of order ≤ 0.


Holomorphic Function Conjugacy Class Order Zero Distribution Solution Regular System 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Andronikof, Microlocalisation tempérée, Mém. Soc. Math. France 57 (1994), Supl. au Bull. de la Soc. Math. France 122 (2).Google Scholar
  2. 2.
    E. Andronikof, A conjugacy class of regular operators, Microlocal geometry (Kyoto 1992), RIMS kôkyûroku, Kyoto Univ. 845, 1993, pp. 8–12.Google Scholar
  3. 3.
    E. Andronikof and T. Monteiro Fernandes, On the tempered solutions of regular systems, Proc. Intern. Conf. “D-modules and microlocal geometry” Lisbon (1990), de Gruyter, Berlin, New-York.Google Scholar
  4. 4.
    T. Aoki, Calcul exponentiel des opérateurs microdifférentiels d’ordre infini, Ann. Inst. Fourier, Grenoble, 33(4) (1983), 227–250.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    M. Kashiwara and T. Oshima, Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math. 106 (1977), 145–200.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    M. Sato, T. Kawai, and M. Kashiwara, Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations (H. Komatsu, ed.), Lecture Notes in Math. 287, Springer, 1973, Proceedings Katata 1971, pp. 265–529.Google Scholar

Copyright information

© Springer-Verlag Tokyo 1997

Authors and Affiliations

  • Emmanuel Andronikof

There are no affiliations available

Personalised recommendations