Let
${\mathcal{V}}$
be a complete discrete valuation ring of unequal characteristic with perfect residue field,
$u:{\mathcal{Z}}{\hookrightarrow}\mathfrak{X}$
be a closed immersion of smooth, quasi-compact, separated formal schemes over
${\mathcal{V}}$
,
$T$
be a divisor of
$X$
such that
$U:=T\cap Z$
is a divisor of
$Z$
, and
$\mathfrak{D}$
a strict normal crossing divisor of
$\mathfrak{X}$
such that
$u^{-1}(\mathfrak{D})$
is a strict normal crossing divisor of
${\mathcal{Z}}$
. We pose
$\mathfrak{X}^{\sharp }:=(\mathfrak{X},\mathfrak{D})$
,
${\mathcal{Z}}^{\sharp }:=({\mathcal{Z}},u^{-1}\mathfrak{D})$
and
$u^{\sharp }:{\mathcal{Z}}^{\sharp }{\hookrightarrow}\mathfrak{X}^{\sharp }$
the exact closed immersion of smooth logarithmic formal schemes over
${\mathcal{V}}$
. In Berthelot’s theory of arithmetic
${\mathcal{D}}$
-modules, we work with the inductive system of sheaves of rings
$\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T):=(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T))_{m\in \mathbb{N}}$
, where
$\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}$
is the
$p$
-adic completion of the ring of differential operators of level
$m$
over
$\mathfrak{X}^{\sharp }$
and where
$T$
means that we add overconvergent singularities along the divisor
$T$
. Moreover, Berthelot introduced the sheaf
${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}:=\underset{\underset{m}{\longrightarrow }}{\lim }\,\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(m)}(T)\otimes _{\mathbb{Z}}\mathbb{Q}$
of differential operators over
$\mathfrak{X}^{\sharp }$
of finite level with overconvergent singularities along
$T$
. Let
${\mathcal{E}}^{(\bullet )}\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{\mathfrak{X}^{\sharp }}^{(\bullet )}(T))$
and
${\mathcal{E}}:=\varinjlim ~({\mathcal{E}}^{(\bullet )})$
be the corresponding object of
$D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}})$
. In this paper, we study sufficient conditions on
${\mathcal{E}}$
so that if
$u^{\sharp !}({\mathcal{E}})\in D_{\text{coh}}^{\text{b}}({\mathcal{D}}_{{\mathcal{Z}}^{\sharp }}^{\dagger }(\text{}^{\dagger }U)_{\mathbb{Q}})$
then
$u^{\sharp (\bullet )!}({\mathcal{E}}^{(\bullet )})\in \underset{\displaystyle \longrightarrow }{LD}\text{}_{\mathbb{Q},\text{coh}}^{\text{b}}(\widehat{{\mathcal{D}}}_{{\mathcal{Z}}^{\sharp }}^{(\bullet )}(U))$
. For instance, we check that this is the case when
${\mathcal{E}}$
is a coherent
${\mathcal{D}}_{\mathfrak{X}^{\sharp }}^{\dagger }(\text{}^{\dagger }T)_{\mathbb{Q}}$
-module such that the cohomological spaces of
$u^{\sharp !}({\mathcal{E}})$
are isocrystals on
${\mathcal{Z}}^{\sharp }$
overconvergent along
$U$
.