We study the nonlinear Schr\"odinger 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 Schr\"odinger 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.
Categories: math.AP