  In our previous paper, given an integral log smooth morphism $X\to S$ of fine
log schemes of characteristic $p>0$, we studied the Azumaya nature of the sheaf
of log differential operators of higher level and constructed a splitting
module of it under an existence of a certain lifting modulo $p^{2}$.
  In this paper, under a certain liftability assumption which is stronger than
our previous paper, we construct another splitting module of our Azumaya
algebra over a scalar extension, which is smaller than our previous paper. As
an application, we construct an equivalence, which we call the log local
Cartier transform of higher level, between certain $\cal D$-modules and certain
Higgs modules. We also discuss about the compatibility of the log Frobenius
descent and the log local Cartier transform and the relation between the
splitting module constructed in this paper and that constructed in the previous
paper. Our result can be considered as a generalization of the result of
Ogus-Vologodsky, Gros-Le Stum-Quir\'os to the case of log schemes and that of
Schepler to the case of higher level.
