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

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.