Microdifferential operator


In mathematics, a microdifferential operator is a linear operator on a cotangent bundle that generalizes a differential operator and appears in the framework of microlocal analysis as well as in the Kyoto school of algebraic analysis.
The notion was originally introduced by L. Boutet de Monvel and P. Krée as well as by M. Sato, T. Kawai and M. Kashiwara. There is also an approach due to J. Sjöstrand.

Definition

We first define the sheaf of formal microdifferential operators on the cotangent bundle of an open subset. A section of that sheaf over an open subset is a formal series: for some integer m,
where each is a holomorphic function on that is homogeneous of degree in the second variable.
The sheaf of microdifferential operators on is then the subsheaf of consisting of those sections satisfying the growth condition on the negative terms; namely, for each compact subset, there exists an such that

Works

  • Aoki, T., Calcul exponentiel des opérateurs microdifférentiels d'ordre infini, I, Ann. Inst. Fourier, Grenoble, 33–4, 227–250.
  • Boutet De Monvel, Louis ; Krée, Paul, Pseudo-differential operators and Gevrey classes, Annales de l'Institut Fourier, Volume 17 no. 1, pp. 295-323
  • M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, in: Lecture Notes in Math. 287, Springer, 1973, 265–529.
  • Sjöstrand, Johannes. Singularités analytiques microlocales, dans Singularités analytiques microlocales - équation de Schrödinger et propagation des singularités..., Astérisque, no. 95, pp. iii-166. https://www.numdam.org/item/AST_1982__95__R3_0/