  We geometrize the mod $p$ Satake isomorphism of Herzig and
Henniart-Vign\'eras using Witt vector affine flag varieties for reductive
groups in mixed characteristic. We deduce this as a special case of a formula,
stated in terms of the geometry of generalized Mirkovi\'c-Vilonen cycles, for
the Satake transform of an arbitrary pararhoric mod $p$ Hecke algebra with
respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula
for the convolution product in an arbitrary parahoric mod $p$ Hecke algebra.
Our methods involve the constant term functors inspired from the geometric
Langlands program, and we also treat the case of reductive groups in equal
characteristic. We expect this to be a first step towards a geometrization of a
mod $p$ Local Langlands Correspondence.

Number Theory;Representation Theory