Microdifferential operator

Definition

We first define the sheaf ${\displaystyle {\widehat {\mathcal {E}}}}$  of formal microdifferential operators on the cotangent bundle ${\displaystyle T^{*}X}$  of an open subset ${\displaystyle X\subset \mathbb {C} ^{n}}$ . A section of that sheaf over an open subset ${\displaystyle U\subset T~*X}$  is a formal series:

Failed to parse (syntax error): {\displaystyle P = \sum_{\infty < j \le m p_j}

where ${\displaystyle p_{j}}$  is a holomorphic function on ${\displaystyle U}$  that is homogeneous of degree $j$ in the second variable.

Reference

• Schapira, Pierre (1985). Microdifferential Systems in the Complex Domain. Grundlehren der mathematischen Wissenschaften. Vol. 269. Springer. doi:10.1007/978-3-642-61665-5. ISBN 978-3-642-64904-2.

Further reading

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