Remarks on the well-posedness of the energy-critical inhomogeneous Hartree equation

Abstract: We study the energy-critical inhomogeneous Hartree equation in space dimensions three and higher. Previous local well-posedness results left open the parameter regime where the inhomogeneity exponent is small and the Riesz potential exponent is either small or large. We establish local well-posedness for a new range of parameters, thereby substantially filling the remaining open parameter regime. In particular, our result completely resolves the remaining gap in dimensions $5$ and $6$.