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We establish some local and global well-posedness for Hartree-Fock equationsof $N$ 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 inFourier-Lebesgue spaces $ \hat{L}^p \cap L^2 \ (1\leq p \leq \infty).$ On theother hand, we show that the Cauchy problem for HFP is ill-posed if we simplywork 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 forHartree type non linearity in these spaces which may be of independentinterest. As a consequence, we get natural $L^p$ and $\hat{L}^p$ extension ofclassical well-posedness theories of Hartree and Hartree-Fock equations withCauchy data in just $L^2-$based Sobolev spaces.

The Hartree and Hartree-Fock equations in Lebesgue $L^p$ and Fourier-Lebesgue $\hat{L}^p$ spaces