  Curves in positive characteristic have a Cartier operator acting on their
space of regular differentials. The $a$-number of a curve is defined to be the
dimension of the kernel of the Cartier operator. In \cite{BoCaASc}, Booher and
Cais used a sheaf-theoretic approach to give bounds on the $a$-numbers of
Artin-Schreier covers. In this paper, that approach is generalized to arbitrary
powers of the Cartier operator, yielding bounds for the dimension of the
kernel. These bounds give new restrictions on the Ekedahl-Oort type of
Artin-Schreier covers.
