On the nonlinear Schrodinger equation in spaces of infinite mass and low regularity

We study the nonlinear Schrodinger equation with initial data in $mathcal{Z}^s_p(mathbb{R}^d)=dot{H}^s(mathbb{R}^d)cap L^p(mathbb{R}^d)$, where $0<s<min{d/2,1}$ and $2<p<2d/(d-2s)$. After showing that the linear Schrodinger group is well-defined in this space, we prove local well-posedness in the whole range of parameters $s$ and $p$. The precise properties of the solution depend on the relation between the power of the nonlinearity and the integrability $p$. Finally, we present a global existence result for the defocusing cubic equation in dimension three for initial data with infinite mass and energy, using a variant of the Fourier truncation method.