Algebraic analysis

It helps in the simplification of the proofs due to an algebraic description of the problem considered.

Microfunction

Let M be a real-analytic manifold of dimension n, and let X be its complexification. The sheaf of microlocal functions on M is given as

${\displaystyle {\mathcal{H}}^n({\mu }_M({\mathcal{O}}_X)\otimes {\mathcal{o}\mathcal{r}}_{M/X})}$

where

• ${\displaystyle {\mu }_M}$ denotes the microlocalization functor,
• ${\displaystyle {\mathcal{o}\mathcal{r}}_{M/X}}$ is the relative orientation sheaf.

A microfunction can be used to define a Sato's hyperfunction. By definition, the sheaf of Sato's hyperfunctions on M is the restriction of the sheaf of microfunctions to M, in parallel to the fact the sheaf of real-analytic functions on M is the restriction of the sheaf of holomorphic functions on X to M.