The inversion formulae for automorphisms of polynomial algebras and rings of differential operators in prime characteristic

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra Pn≔K[x1,…,xn], the ring of differential operators D(Pn) on Pn, D(Pn)⊗Pm, the n'th Weyl algebra An or An⊗Pm, the power series algebra K[[x1,…,xn]], the subalgebra Tk1,…,kn of D(Pn) generated by Pn and the higher derivations ∂i[j], 0≤j