  Berglund-H\"ubsch duality is an example of mirror symmetry between orbifold
Landau-Ginzburg models. In this paper we study a D-module-theoretic variant of
Borisov's proof of Berglund-H\"ubsch duality. In the $p$-adic case, the
D-module approach makes it possible to endow the orbifold chiral rings with the
action of a non-trivial Frobenius endomorphism. Our main result is that the
Frobenius endomorphism commutes with Berglund-H\"ubsch duality up to an
explicit diagonal operator.
