Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, and I can ask that a sheaf be constant after pulling back to the orbits of this action (i.e. rays in the fibers of $E$). Sheaves that satisfy this property are called conic sheaves.
There is an operation we can define which takes conic sheaves on $E$ to conic sheaves on the dual bundle $E^*$ called the Fourier-Sato transform, defined as follows: Define $P' = \{(x,y) \in E \times_M E^*: \langle x,y \rangle \leq 0\}$. Given a conic sheaf $G$ we define the Fourier-Sato transform to be $\hat{G} = R\pi^*_!(\pi^{-1}G)_{P'}$, where $\pi^*$ denotes the projection map of the dual bundle.
Now given a sheaf $F$ on an arbitrary manifold $M$, we can define the specialization of $F$ with respect to a given submanifold $N$, which is a conic sheaf on the normal bundle to $N$, denoted by $\nu_N(F)$. The specialization functor is defined by first forming the normal deformation of $N$ in $M-$a sort of a "real blow-up" of the submanifold $N-$ creating a family over $\mathbb{R}$ equipped with a map to $M$ and whose fiber at $0$ is the normal bundle $T_NM$; and then doing some push-pull operations akin to the nearby cycles functor.
Putting the specialization functor $\nu_N$ and the Fourier-Sato transform together gives us the microlocalization functor: $\mu_N(F) = \nu_N(F)$^.
At this point, I have an okay understanding of the specialization functor as something that takes a sheaf $F$ and asks how it behaves infinitesimally close to $N$ in the normal directions to $N$, like we're kind of taking dough and stretching it thin in only normal directions, then asking what our sheaf looks like in this stretched picture. I also have an okay understanding of the Fourier-Sato transform as being defined the way it is to imitate the classical Fourier transform by switching skyscraper and constant sheaves and perhaps have restriction to $P'$ play the role of $e^{-ix \cdot \xi}$.
My question is: how can I see that the microlocalization functor somehow organizes or is related to the microlocal information of my sheaf like its name suggests? I'd appreciate at least some pseudo-geometric intuition in the style of what I've written above Moreover, is my intuition as written above good or misleading?
These definitions are all contained in chapters 3 and 4 of Sheaves on Manifolds by Kashiwara, Schapira, but going through their proofs often leaves me, at best, unsure of the ideas behind the proof, and, at worst, hopelessly confused even if I follow individual steps in the proofs. Any help?
I'm including the algebraic geometry tag on this because I'm aware there are related constructions in the algebraic world.
