  Let $\mathfrak{X}$ be a formal smooth quasi-compact scheme over a complete
discrete valuation ring $\mathcal{V}$ of mixed characteristic $(0 , p)$.We
consider the sheaves of differential operators
$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ with a congruence
level $k \in \mathbb{N}$ and their projective limit $\mathcal{D}_{\mathfrak{X},
\infty} = \varprojlim_k \widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k ,
\mathbb{Q}}$.In this article, we construct microlocalization sheaves of
$\widehat{\mathcal{D}}^{(0)}_{\mathfrak{X}, k , \mathbb{Q}}$ for the different
congruence levels $k$ admitting transition morphisms.Then we pass to the
projective limit over $k$ in order to obtain some microlocalization sheaves of
$\mathcal{D}_{\mathfrak{X}, \infty}$.These sheaves will be useful later to
define a characteristic variety for coadmissible $\mathcal{D}_{\mathfrak{X},
\infty}$-modules when $\mathfrak{X}$ is a formal curve.
