Local Cauchy theory for the nonlinear Schr\"odinger equation in spaces of infinite mass

We consider the Cauchy problem for the nonlinear Schr"odinger equation on ${\mathbb{R}}^d$, where the initial data is in $\dot{H}^1($\mathbb{R}$^d)\cap L^p($\mathbb{R}$^d)$\mathrm{.}Weprovelocalwell-posednessforlargerangesof$p$ and discuss some global well-posedness results.