Algebraic analysis

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[2]

${\displaystyle {\mathcal {H}}^{n}(\mu _{M}({\mathcal {O}}_{X})\otimes {\mathcal {or}}_{M/X})}$

where

• ${\displaystyle \mu _{M}}$ denotes the microlocalization functor,
• ${\displaystyle {\mathcal {or}}_{M/X}}$ 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.

• Hyperfunction
• D-module
• Microlocal analysis
• Generalized function
• Edge-of-the-wedge theorem
• FBI transform
• Localization of a ring
• Vanishing cycle
• Gauss–Manin connection
• Differential algebra
• Perverse sheaf
• Mikio Sato
• Masaki Kashiwara
• Lars Hörmander

• Kashiwara, Masaki; Schapira, Pierre (1990). Sheaves on Manifolds. Berlin: Springer-Verlag. ISBN 3-540-51861-4.

• Masaki Kashiwara and Algebraic Analysis
• Foundations of algebraic analysis book review