My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in this paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a $\mathcal{D}$-module of level $m$ is a module over $\mathbb{k} \langle x, \partial_x, \frac{{\partial_x}^p}{p!}, \cdots, \frac{{\partial_x}^{p^m}}{(p^m)!} \rangle$; I was wondering how to work this out from first principles, using Definition $2.5$ given in that paper.

Consider the immersion $X \rightarrow X \times_S X$. Definition $2.1$ from that paper defines what a divided power structure of level $m$ on an immersion is; Definition $2.3$ (and $2.4$) states constructs the divided power envelope of level $m$, $P_{X,m}(Y)$. Subsequently, they define $\mathcal{P}_{X,m}^n(Y)$, whose dual is defined to be $\mathcal{D}^m_{X,n}$. Most of the details/proofs are done in this other paper.

I'm having trouble properly understanding these definitions and working out what they are in the case of $X=\mathbb{A}^1, S=\text{Spec } \overline{\mathbb{F}_p}$. So, my question is what the above objects look like in this particular example, and how to explicitly calculate everything in this example.

My question is regarding the definition of $\mathcal{D}$-modules of level $m$ given in this paper. As an example, let $X=\mathbb{A}^1$ over $S=\text{Spec }\overline{\mathbb{F}_p}$; I was told that a $\mathcal{D}$-module of level $m$ is a module over $\mathbb{k} \langle x, \partial_x, \frac{{\partial_x}^p}{p!}, \cdots, \frac{{\partial_x}^{p^m}}{(p^m)!} \rangle$; I was wondering how to work this out from first principles, using Definition $2.5$ given in that paper.

Consider the immersion $X \rightarrow X \times_S X$. Definition $2.1$ from that paper defines what a divided power structure of level $m$ on an immersion is; Definition $2.3$ (and $2.4$) states constructs the divided power envelope of level $m$, $P_{X,m}(Y)$. Subsequently, on pg $5$ they define $\mathcal{P}_{X,m}^n(Y)$; the sheaf of differential operators of level $m$ is defined to be union of the duals of this family of sheaves (as $n$ varies). Most of the details/proofs are done in this other paper.

I'm having trouble properly understanding these definitions and working out what they are in the case of $X=\mathbb{A}^1, S=\text{Spec } \overline{\mathbb{F}_p}$. So, my question is what the above objects look like in this particular example, and how to explicitly calculate everything in this example.