Powers of the Cartier operator on Artin–Schreier covers

For a curve in positive characteristic, the Cartier operator acts on the vector space of its regular differentials. The a-number is defined to be the dimension of the kernel of the Cartier operator. In a-numbers of curves in Artin–Schreier covers, Algebra Number Theory 14(3) (2020) 587–641, Booher and Cais use a sheaf-theoretic approach to give bounds on the a-numbers of Artin–Schreier covers. In this paper, I generalize that approach 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.