Rings of microdifferential operators for arithmetic $\mathcal{D}$-modules

The aim of this paper is to develop a theory of microdifferential operators for arithmetic $\mathcal{D}$-modules. We first define the sheaves of microdifferential operators of arbitrary levels on arbitrary smooth formal schemes. A difficulty lies in the fact that there are no homomorphisms between sheaves of microdifferential operators of different levels. To remedy this, we define the intermediate differential operators, and using these, we define the sheaf of microdifferential operators for ${\mathcal{D}}^{\mathrm{†}}$. We conjecture that the characteristic variety of a ${\mathcal{D}}^{\mathrm{†}}$-module is computed as the support of the microlocalization of a ${\mathcal{D}}^{\mathrm{†}}$-module, and prove it in the curve case.