Nonlinear Resonances with a Potential: Multilinear Estimates and an Application to NLS

Abstract

This paper considers the question of global in time existence and asymptotic behavior of small-data solutions of nonlinear dispersive equations with a real potential |$V$|⁠. The main concern is treating nonlinearities whose degree is low enough as to preclude the simple use of classical energy methods and decay estimates. In their place, we present a systematic approach that adapts the space-time resonance method to the non-Euclidean setting using the spectral theory of the Schrödinger operator |$-\Delta +V$|⁠. We start by developing tools of independent interest, namely multilinear analysis (Coifman–Meyer type theorems) in the framework of the corresponding distorted Fourier transform. As a first application, this is then used to prove global existence and scattering for a quadratic Schrödinger equation.