Constant term functors with Fₚ-coefficients

We study the constant term functor for Fₚ -sheaves on the affine Grassmannian in characteristic p with respect to a Levi subgroup. Our main result is that the constant term functor induces a tensor functor between categories of equivariant perverse Fₚ -sheaves. We apply this fact to get information about the Tannakian monoids of the corresponding categories of perverse sheaves. As a byproduct we also obtain geometric proofs of several results due to Herzig on the mod p Satake transform and the structure of the space of mod p Satake parameters.