In characteristic $p > 0$ there are "extra" differential operators, i.e., ones that are outside the algebra generated by first-order derivations.

Is there any interpretation of these operators in terms of Grothendieck (or other) style geometric ideas such as thickenings, formal schemes, lambda rings, or crystalline cohomology?

A related and possibly equivalent question is whether there is any precise sense in which the "extra" material in divided power algebras (compared to polynomial or power-series rings) is dual to the "extra" differential operators in positive characteristic(s).

definedin terms of thickenings: The sheaf of operators of order <= n on X is dual to the structure sheaf of the n-th infinitesimal neighborhood of the diagonal of X in X x X. Similarly, the sheaf of all diff ops is dual to the structure sheaf of the formal nbhd. of the diagonal. For crystalline theory you add divided power structures, which make the extra operators go away. Ring of PD-DiffOps=ring gen'd by derivations and O_X if I remember correctly. – Lars Jun 24 '10 at 6:08