Microlocalization is a tool invented by Mikio Sato to study linear partial differential equations (as a part of his algebraic analysis program) not only locally in space but also locally in momentum variable. It is a purely algebraic theory that was also continued in parallel by analysts, like Hormander, giving the domain of microlocal analysis.
The original construction is based on the use of the specialization functor and Fourier-Sato transformation. In this section, we will discuss the construction in a general setting, i.e., over an arbitrary field of characteristic . In the real situation, one usually refines the construction by using conic sheaves (i.e., sheaves invariant with respect to the natural -action) to get information about the oriented direction of propagation of singularities of sheaves of solutions of analytic partial differential systems. The construction we describe does not treat this refined information.
Let be a closed subspace of a given analytic manifold, defined by a sheaf of ideals , with normal bundle denoted and conormal bundle denoted . One defines the deformation to the normal bundle as the (analytic space associated to the) relative scheme over given by
with for . There is a projection and a projection . The fiber at of is denoted , and its fiber at is . The fiber of on the open subset is .
The specialization of a sheaf is the sheaf defined as
The Fourier-Sato transform is the functor
where and are the two natural projections.
The -microlocalization functor is the functor
The microlocalization functor on a variety is defined as the -microlocalization associated to the closed immersion . Since , this gives a functor
Denoting the natural projection, we defined the microlocal homomorphisms between two complexes of sheaves on on by
If is the natural projection, we have
Sato’s theory of microlocalization was first described in the setting of D-modules:
This theory of microlocalization of (ind)-sheaves (and also sub-analytic sheaves) was developped in the following works:
A good overview of the theory can by found at: