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

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