Bi- and trilinear Schroedinger 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 Schroedinger equation in one space dimension are estimated in mixed and, in the first case weighted, L^p - 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 Schroedinger (NLS) equation with data u_0 in the function space ^L^r:=^H^r_0, where for s \in R the spaces ^H^r_s are defined by the norms ||u_0||_{^H^r_s}:=||^u_0||_{L^r&#39;_ξ}, 1/r + 1/r&#39;=1. Similar arguments, combined with a gauge transform, lead to local well-posedness of the Cauchy problem for the derivative nonlinear Schroedinger (DNLS) equation with data u_0 \in ^H^r_{1/2}. In the local result on cubic NLS the parameter r is allowed in the whole subcritical range 1<r<\infty, while for DNLS we assume 1<r \le 2. In the special case r=2 both results coincide with the optimal ones on the H^s - scale. Furthermore, concerning the cubic NLS equation, it is shown by a decomposition argument that the local solution extends globally, provided 2 \ge r > 5/3.