  In this paper we obtain Sobolev estimates for weak solutions of first oder
variational Mean Field Game systems with coupling terms that are local function
of the density variable. Under some coercivity condition on the coupling, we
obtain first order Sobolev estimates for the density variable, while under
similar coercivity condition on the Hamiltonian we obtain second order Sobolev
estimates for the value function. These results are valid both for stationary
and time-dependent problems. In the latter case the estimates are fully global
in time, thus we resolve a question which was left open in [PS17]. Our methods
apply to a large class of Hamiltonians and coupling functions.
