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

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