Let
$\mathfrak{o}$
be a complete discrete valuation ring of mixed characteristic
$(0,p)$
and
$\mathfrak{X}_{0}$
a smooth formal
$\mathfrak{o}$
-scheme. Let
$\mathfrak{X}\rightarrow \mathfrak{X}_{0}$
be an admissible blow-up. In the first part, we introduce sheaves of differential operators
$\mathscr{D}_{\mathfrak{X},k}^{\dagger }$
on
$\mathfrak{X}$
, for every sufficiently large positive integer
$k$
, generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme
$\mathfrak{X}_{0}$
. The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf
$\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$
and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves
$\mathscr{D}_{\mathfrak{X},\infty }$
, over all admissible blow-ups
$\mathfrak{X}$
, is a sheaf
$\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$
on the Zariski–Riemann space of
$\mathfrak{X}_{0}$
, which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for
$\mathscr{D}_{\mathfrak{X},k}^{\dagger }$
,
$\mathscr{D}_{\mathfrak{X},\infty }$
, and
$\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$
.