Global well-posedness for NLS outside L^2
Abstract：We first introduce a new function space whose norm is given by the l^p-sum of modulated Sobolev norms of a given function. In particular, we show that this space agrees with the modulation space on the real line and the Fourier-Lebesgue space on the circle. We use this equivalence of the norms and the Galilean symmetry to adapt the conserved quantities constructed by Killip-Visan-Zhang to the modulation space and Fourier-Lebesgue space setting. By applying the scaling symmetry, we then prove global well-posedness of the one-dimensional cubic nonlinear Schroedinger equation (NLS) in almost critical spaces. More precisely, we show that the cubic NLS on the real line is globally well-posed in almost critical modulation spaces, while the renormalized cubic NLS on Torus is globally well-posed in almost critical Fourier-Lebesgue space. This is a joint work with Tadahiro Oh at the University of Edinburgh.