  In order to define a geometric Fourier transform, one usually works with
either $\ell$-adic sheaves in characteristic $p>0$ or with $D$-modules in
characteristic 0. If one considers $\ell$-adic sheaves on the stack quotient of
a vector bundle $V$ by the homothety action of $\mathbb{G}_m$, however, Laumon
provides a uniform geometric construction of the Fourier transform in any
characteristic. The category of sheaves on $[V/\mathbb{G}_m]$ is closely
related to the category of (unipotently) monodromic sheaves on $V$. In this
article, we introduce a new functor, which is defined on all sheaves on $V$ in
any characteristic, and we show that it restricts to an equivalence on
monodromic sheaves. We also discuss the relation between this new functor and
Laumon's homogeneous transform, the Fourier-Deligne transform, and the usual
Fourier transform on $D$-modules (when the latter are defined).
