Microdifferential operator

In mathematics, a microdiiferential operator is a linear operator on a cotangent bundle (phase space) that generalizes a differential operator and appears in the framework of microlocal analysis.

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 ${\displaystyle X\subset \mathbb {C} ^{n}}$ .^[1] A section of that sheaf over an open subset ${\displaystyle U\subset T^{*}X}$  is a formal series: for some integer m,

${\displaystyle P=\sum _{-\infty <j\leq m}p_{j}}$ 

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

The sheaf ${\displaystyle {\mathcal {E}}}$  of microdifferential operators on ${\displaystyle T^{*}X}$  is then the subsheaf of ${\displaystyle {\widehat {\mathcal {E}}}}$  consisting of those satisfying the growh condition on the negative terms; namely, for each compact subset ${\displaystyle K\subset U}$ , there exists an ${\displaystyle \epsilon >0}$  such that

${\displaystyle \sum _{j\leq 0}\sup _{K}|p_{j}|\epsilon ^{-j}/(-j)!<\infty .}$ ^[2]

See also

• pseudodifferential operator

Reference

Notes

1. Schapira 1985, Ch. I., § 1.2.
2. Schapira 1985, Ch. I., § 1.3.

Works

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