Fourier transform and exponential sheaves

This note concerns exponential sheaves and the "universal" Fourier transform on them. Fourier invertibility and the subsequent Fourier miracle is demonstrated. Further, t-structures and realizations are constructed and shown to have favorable properties. In particular, the Fourier transform constructed is shown to commute, under realizations, with its classical counterparts (whenever the latter exist). The motivation is to understand the "analogies" between exponential sums over finite fields and differential equations in the sense of N. Katz's works.