Modified Scattering for the Hartree Nonlinear Schrödinger Equation

We prove sharp $L^{\mathrm{\infty }}$ decay and modified scattering for the Hartree nonlinear Schrödinger equation in dimensions $2$ and $3$ using the testing by wavepackets method of Ifrim and Tataru. We show that the scattering behavior happens at a regularity well below that of earlier results of Hayashi-Naumkin and Kato-Pusateri.