Microdifferential operator: Difference between revisions

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In mathematics, a '''microdifferential operator''' is a linear operator on a cotangent bundle (phase space) 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<ref>{{harvnb|L. Boutet De Monvel, Louis|P. Krée}}</ref> as well as by M. Sato, T. Kawai and M. Kashiwara.<ref>{{harvnb|M. Sato|T. Kawai|M. Kashiwara}}</ref> There is also an approach due to J. Sjöstrand.<ref>{{harvnb| Sjöstrand}}</ref>

== Definition ==

We first define the sheaf <math>\widehat{\mathcal{E}}</math> of formal microdifferential operators on the cotangent bundle <math>T^* X</math> of an open subset <math>X \subset \mathbb{C}^n</math>.<ref>{{harvnb|Schapira|1985|loc=Ch. I., § 1.2.}}</ref> A section of that sheaf over an open subset <math>U \subset T^* X</math> is a formal series: for some integer ''m'',

:<math>P = \sum_{-\infty < j \le m} p_j</math>

where each <math>p_j</math> is a [[holomorphic function]] on <math>U</math> that is [[homogeneous function|homogeneous]] of degree <math>j</math> in the second variable.

The sheaf <math>\mathcal{E}</math> of microdifferential operators on <math>T^* X</math> is then athe formalsubsheaf microdifferentialof operator<math>\widehat{\mathcal{E}}</math> thatconsisting satisfiesof those secctions satisfying the growh condition on the negative terms; namely, for each compact subset <math>K \subset U</math>, there exists an <math>\epsilon > 0</math> such that

:<math>\sum_{j \le 0} \sup_K|p_j| \epsilon^{-j}/(-j)! < \infty.</math><ref>{{harvnb|Schapira|1985|loc=Ch. I., § 1.3.}}</ref>

== See also ==

*[[Pseudodifferential operator]]

<!--*[[quantized contact transformation]], this article doesn't exist yet.-->

== Reference ==

===Notes===

{{reflist}}

===Works===

* Aoki, T., Calcul exponentiel des opérateurs microdifférentiels d'ordre infini, I, Ann. Inst. Fourier, Grenoble, 33–4 (1983), 227–250.

* Boutet De Monvel, Louis ; Krée, Paul, Pseudo-differential operators and Gevrey classes, Annales de l'Institut Fourier, Volume 17 (1967) 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.

*{{cite book |last1=Schapira |first1=Pierre |title=Microdifferential Systems in the Complex Domain |series=Grundlehren der mathematischen Wissenschaften |date=1985 |volume=269 |publisher=Springer |doi=10.1007/978-3-642-61665-5 |isbn=978-3-642-64904-2 |url=https://link.springer.com/book/10.1007/978-3-642-61665-5}}

* Sjöstrand, Johannes. Singularités analytiques microlocales, dans Singularités analytiques microlocales - équation de Schrödinger et propagation des singularités..., Astérisque, no. 95 (1982), pp. iii-166. https://www.numdam.org/item/AST_1982__95__R3_0/

== Further reading ==

* https://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1723-09.pdf in Japanese

[[Category:Differential operators]]