Modified Scattering for Nonlocal Nonlinear Schrödinger Equations

We prove a modified scattering and sharp $L^{\mathrm{\infty }}$ decay result for both the Hartree and Schrödinger-Bopp-Podolsky equations in dimensions $2$ and $3$ using the testing by wavepackets approach due to Ifrim and Tataru. We show that modified scattering and sharp pointwise decay occur for these equations at a regularity much lower than previous results due to Hayashi-Naumkin and Kato-Pusateri, and as a corollary also show that the results on power-type scattering-critical NLS due to Hayashi-Naumkin can be proven under minimal regularity assumptions.