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

We consider the initial value problem associated to the inhomogeneous nonlinear Schrodinger equation, iu(t )+ Delta u + mu|x|(-b)|u|(alpha)u = 0, u(0 )is an element of H-s(R-N) or u(0 )is an element of H-s(R-N), with mu = +/- 1, b > 0, s >= 0 and 0 < alpha <= 4-2b/N-2s (0 < alpha < infinity if s >= 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.