L^p Boundedness of the Scattering Wave Operators of Schroedinger Dynamics with Time-dependent Potentials and applications

This paper establishes the $L^p$ boundedness of wave operators for linear Schr"odinger equations in ${\mathbb{R}}^3$ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in $L^{\mathrm{\infty }}$, for a class of Hartree nonlinear Schr"odinger equations in $L^2({\mathbb{R}}^3),$ allowing the presence of solitons. We also prove the existence of free channel wave operators in $L^p({\mathbb{R}}^n)$ for $p>p_c(n)$, with $p_c(3)=6$.