  We study the Cauchy problem for the inhomogeneous Hartree equation in this
paper. Although its well-posedness theory has been extensively studied in
recent years, much less is known compared to the classical Hartree model of
homogeneous type. In particular, the problem of Sobolev initial data with the
Sobolev critical index remains unsolved. The main contribution of this paper is
to establish the local existence of solutions to the inhomogeneous equation in
the critical cases. To do so, we obtain all possible $L^p$ Strichartz estimates
with singular weights.
