Calculation of l-adic local Fourier transformations

We calculate the local Fourier transformations for a class of $\bar{\mathbb Q}_\ell$-sheaves. In particular, we verify a conjecture of Laumon and Malgrange. As an application, we calculate the local monodromy of $\ell$-adic hypergeometric sheaves introduced by Katz. We also discuss the characteristic $p$ analogue of the Turrittin-Levelt Theorem for $D$-modules. The method used in this paper can be used to show a conjecture of Ramero which states that the Fourier transformation of an analytic sheaf with meromorphic ramification still has meromorphic ramification.