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

We consider the initial value problem associated to the inhomogeneous nonlinear Schrödinger equation, $i{u}_t+\Delta u+\mu {\left|x\right|}^{-b}{\left|u\right|}^{\alpha }u=0,u_0\in H^s\left(R^N\right)\text{or}{u}_0\in {\dot{H}}^s\left(R^N\right),$with $\mu =\pm 1$, $b>0$, $s\ge 0$ and $0<\alpha \le \frac{4-2b}{N-2s}$ ($0<\alpha <\infty$ if $s\ge N/2$). 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.