Microdifferential operator: Difference between revisions

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In mathematics,

== 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'',

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The sheaf <math>\mathcal{E}</math> of microdifferential operators on <math>T^* X</math> is then a formal microdifferential operator that satisfies 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 ==

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

*[[psuedodifferential operator]]

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

== Reference ==