D-modules arithm\'etiques associ\'es aux isocristaux surconvergents. Cas lisse

Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $\mathcal{P}$ a separated smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $X$ a smooth closed subscheme of $P$, $T$ a divisor in $P$ such that $T_X=T\cap X$ is a divisor in $X$ and $\smash{\D}^\dag {$\mathcal{P}$}(\hdag T)$ the weak completion of the sheaf of differential operators on $\mathcal{P}$ with overconvergent singularities along $T$. We construct a fully faithful functor denoted by $ \sp{X \hookrightarrow $\mathcal{P}$,T,+}$fromthecategoryofisocrystalon$X \setminus T_X$overconvergentalong$T_X$intothecategoryofcoherent$\smash{\D}^\dag {$\mathcal{P}$}(\hdag T) \otimes$\mathbb{Z}$ $\mathbb{Q}$ $-moduleswithsupportin$X$\mathrm{.}Next,weprovethecommutationof$ \sp_{X \hookrightarrow $\mathcal{P}$,T,+}$ with (extraordinary) inverse images and dual functors. These properties are compatible with Frobenius.