  Given a natural number $m$ and a log smooth integral morphism $X\to S$ of
fine log schemes of characteristic $p>0$ with a lifting of its Frobenius
pull-back $X'\to S$ modulo $p^{2}$, we use indexed algebras ${\cal
A}_{X}^{gp}$, ${\cal B}_{X/S}^{(m+1)}$ of Lorenzon-Montagnon and the sheaf
${\cal D}_{X/S}^{(m)}$ of log differential operators of level $m$ of
Berthelot-Montagnon to construct an equivalence between the category of certain
indexed ${\cal A}_{X}^{gp}$-modules with ${\cal D}_{X/S}^{(m)}$-action and the
category of certain indexed ${\cal B}_{X/S}^{(m+1)}$-modules with Higgs field.
Our result is regarded as a level $m$ version of some results of
Ogus-Vologodsky and Schepler.
