  In this paper, local well-posedness is shown for the one dimensional cubic
nonlinear Schr\"odinger equation in $L^p$-spaces for $2<p<4$, which generalizes
a classical result for $p=2$ by Y. Tsutsumi and recent work for $1<p<2$ by Y.
Zhou. As a consequence, a local theory of solutions is established for a class
of data which decay more slowly than square integrable functions. Regularity
properties of the local solutions in the $L^p$-based Sobolev spaces and
Stricharz spaces are also proved.
