The group of order preserving automorphisms of the ring of differential operators on a Laurent polynomial algebra in prime characteristic

Let $K$ be a field of characteristic $p>0$. It is proved that the group $\mathrm{Aut}_{ord}(\mathcal{D}(L_n))$ of order preserving automorphisms of the ring $\mathcal{D}(L_n)$ of differential operators on a Laurent polynomial algebra $L_n:= K[x_1^{\pm 1}, \ldots, x_n^{\pm 1}]$ is isomorphic to a skew direct product of groups ${\mathbb{Z}}_p^n \rtimes \mathrm{Aut}_K(L_n)$, where ${\mathbb{Z}}_p$ is the ring of $p$-adic integers. Moreover, the group $\mathrm{Aut}_{ord}(\mathcal{D}(L_n))$ is found explicitly. Similarly, $\mathrm{Aut}_{ord}(\mathcal{D}(P_n))\simeq \mathrm{Aut}_K(P_n)$, where $P_n: =K[x_1, \ldots, x_n]$ is a polynomial algebra.