The mixed fractional Hartree equations in Fourier amalgam and modulation spaces

We prove local and global well-posedness for mixed fractional Hartree equation and with low regularity Cauchy data in Fourier amalgam $\text{F}W(L^p,{\mathrm{\ell }}^q)$ and modulation $M^{p,q}$ spaces. Similar results also hold for the Hartree equation with harmonic potential in some modulation spaces. Our approach also addresses Hartree-Fock equations of finitely many (but arbitrary large) particles. A key ingredient of our method is to establish trilinear estimates for Hartree non-linearity and the use of Strichartz estimates. As a consequence, we could gain $\text{F}W(L^p,{\mathrm{\ell }}^q)$ and $M^{p,q}-$regularity for all $p,q\in [1,\mathrm{\infty }]\mathrm{.}$ In particular, we extend result of Bhimani-Grillakis-Okoudju \cite{bhimani2020hartree} in $M^{p,q}$ for all $p,q$ and complement known results in Sobolev spaces.