  Using a theory of $F^e$-modules (a natural extension of Lyubeznik's
$F$-module theory), we extend results on Matlis dual of $F$-finite $F$-modules
to $D$-submodules of $F^e$-finite $F^e$-modules and apply these results to
address the Lyubeznik-Yildirim conjecture in mixed characteristic.
