Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic

Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\text{s}\in {\text{Aut}}_K(\text{CDPn})$ of the ring $\text{CDPn}$ of differential operators on a polynomial algebra $P_n=K[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ is {\em uniquely} determined by the elements $\text{s}(x_1),\mathrm{.}\mathrm{.}\mathrm{.},\text{s}(x_n)$, and the set $\Frob (\CDPn)$ of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of $\text{CDPn}$, like $P_n$, is equal to ${\text{Aut}}_K(\text{CDPn})\cdot \text{CF}$ where $\text{CF}$ is the set of all the extensions of the Frobenius from $P_n$ to $\text{CDPn}$ that leave invariant the subalgebra of scalar differential operators. The set $\text{CF}$ is found explicitly, it is large (a typical extension depends on {\em countably} many independent parameters).