The category of enhanced subanalytic sheaves as a tool for Fourier analysis

In this talk, we introduce the category of enhanced subanalytic sheaves on a complex bordered space by using the subanalytic site and the addition of an extra real variable. We also define Grothendieck operations and convolution functors which allow to see this category as a commutative tensor category. Then, we consider a bordered complex vector space and we define enhanced Fourier-Sato functors, which are equivalence of categories. We finally explain how this framework allows to obtain a "device" which produces explicit holomorphic Paley-Wiener-type theorems thanks to cohomological computations.