On well-posedness for inhomogeneous Hartree equations in the critical case

We study the well-posedness for the inhomogeneous Hartree equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=\lambda (I_{\alpha }\ast \mathrm{\mid }\cdot {\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^p)\mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u$ in $H^s$, $s\ge 0$. Until recently, its well-posedness theory has been intensively studied, focusing on solving the problem for the critical index $p=1+\frac{2-2b+\alpha }{n-2s}$ with $0\le s\le 1$, but the case $1\mathrm{/}2\le s\le 1$ is still an open problem. In this paper, we develop the well-posedness theory in this case, especially including the energy-critical case. To this end, we approach to the matter based on the Sobolev-Lorentz space which can lead us to perform a finer analysis for this equation. This is because it makes it possible to control the nonlinearity involving the singularity $\mathrm{\mid }x{\mathrm{\mid }}^{-b}$ as well as the Riesz potential $I_{\alpha }$ more effectively.