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Let $K$ be a field of characteristic $p>0$. It is proved that eachautomorphism $\s \in \Aut_K(\CDPn)$ of the ring $\CDPn$ of differentialoperators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\em uniquely}determined by the elements $\s (x_1), ... ,\s (x_n)$, and the set $\Frob(\CDPn)$ of all the extensions of the Frobenius from certain maximalcommutative polynomial subalgebras of $\CDPn$, like $P_n$, is equal to$\Aut_K(\CDPn) \cdot \CF$ where $\CF$ is the set of all the extensions of theFrobenius from $P_n$ to $\CDPn$ that leave invariant the subalgebra of scalardifferential operators. The set  $\CF$ is found explicitly, it is large (a typical extension depends on {\emcountably} many independent parameters).

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