  We establish some local and global well-posedness for Hartree-Fock equations
of $N$ particles (HFP) with Cauchy data in Lebesgue spaces $L^p \cap L^2 $ for
$1\leq p \leq \infty$. Similar results are proven for fractional HFP in
Fourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the
other hand, we show that the Cauchy problem for HFP is ill-posed if we simply
work in $\hat{L}^p \ (2<p\leq \infty).$ Analogue results hold for reduced HFP.
In the process, we prove the boundedeness of various trilinear estimates for
Hartree type non linearity in these spaces which may be of independent
interest. As a consequence, we get natural $L^p$ and $\hat{L}^p$ extension of
classical well-posedness theories of Hartree and Hartree-Fock equations with
Cauchy data in just $L^2-$based Sobolev spaces.
