  We consider the initial value problem associated to the inhomogeneous
nonlinear Schr\"o\-din\-ger equation,
  \begin{equation}
  iu_t + \Delta u +\mu|x|^{-b}|u|^{\alpha}u=0, \quad u_0\in H^s(\mathbb R^N)
\text{ or } u_0 \in\dot H ^s(\mathbb R^N),
  \end{equation} with $\mu=\pm 1$, $b > 0$, $s\geq 0$ and $0 < \alpha \leq
\frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz
rule, we prove new local well-posedness results in Sobolev spaces for a large
range of parameters. We also prove an ill-posedness result for this equation,
through a delicate analysis of the associated Duhamel operator.
