Local smoothing estimates of fractional Schrödinger equations in $\alpha$-modulation spaces with some applications

We show some new local smoothing estimates of the fractional Schrödinger 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 Schrödinger equations on the line.