  The purpose of this paper is to study well-posedness of the initial value
problem (IVP) for the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$
i u_t +\Delta u+\lambda|x|^{-b}|u|^\alpha u = 0, $$ where $\lambda=\pm 1$ and
$\alpha$, $b>0$.
  We obtain local and global results for initial data in $H^s(\mathbb{R}^N)$,
with $0\leq s\leq 1$. To this end, we use the contraction mapping principle
based on the Strichartz estimates related to the linear problem.
