Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation

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.