  We show some new local smoothing estimates of the fractional Schr\"odinger
equations with initial data in $\alpha$-modulation spaces via decoupling
inequalities. Furthermore, our necessary conditions show that the local
smoothing estimates are sharp in some cases. As applications, the local
smoothing estimates could show some new local well-posedness on modulation
spaces of the fourth-order nonlinear Schr\"odinger equations on the line.
