  The purpose of this paper is to develop a new theory of gauges in mixed
characteristic. Namely, let $k$ be a perfect field of characteristic $p>0$ and
$W(k)$ the $p$-typical Witt vectors. Making use of Berthelot's arithmetic
differential operators, we define for a smooth formal scheme $\mathfrak{X}$
over $W(k)$, a new sheaf of algebras
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ which can be considered a higher
dimensional analogue of the (commutative) Dieudonne ring. Modules over this
sheaf of algebras can be considered the analogue (over $\mathfrak{X}$) of the
gauges of Ekedahl and Fontain-Jannsen. We show that modules over
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$ admit all of the usual
$\mathcal{D}$-module operations, and we prove a robust generalization of
Mazur's theorem in this context. Finally, we show that an integral form of a
mixed Hodge module of geometric origin admits, after a suitable $p$-adic
completion, the structure of a module over
$\widehat{\mathcal{D}}_{\mathfrak{X}}^{(0,1)}$. This allows us to prove a
version of Mazur's theorem for the intersection cohomology and the ordinary
cohomology of an arbitrary quasiprojective variety defined over a number field.
