Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS

Abstract

The Fourier transforms of the products of two respectively three solutions of the free Schrödinger equation in one space dimension are estimated in mixed and, in the first case, weighted Lp-norms. Inserted into an appropriate variant of the Fourier restriction norm method, these estimates serve to prove local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger (NLS) equation with data u0 in the function spaces Lxr^:=H0r^⁠, where for s ∈ ℝ the spaces Hsr^ are defined by their norms ‖u0‖Hsr^:=‖〈ξ〉su0^‖Lξr',1/r+1/r'=1⁠. Similar agruments, combined with a gauge transform, lead to local well-posedness of the Cauchy problem for the derivative nonlinear Schrödinger (DNLS) equation with data u0∈H1/2r^⁠. In the local result on cubic NLS the parameter r is allowed in the full subcritical range 1 < r < ∞, while for DNLS we assume 1<r ≤ 2. In the special case r=2 both results coincide with the optimal ones on the Hs-scale. Furthermore, concerning the cubic NLS equation, it is shown by a decomposition argument that the local solutions extend globally, provided 2 ≥ r > 5/3.