The Hartree-Fock equations in $L^p$ and Fourier-$L^p$ spaces

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 $ \widehat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On the other hand, we show that Cauchy problem for HFP is ill-posed if we simply work in $\widehat{L}^p \ (2