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

TL;DR

This paper establishes new local smoothing estimates for fractional Schrödinger equations within α-modulation spaces, demonstrating sharpness and applying results to local well-posedness of nonlinear Schrödinger equations.

Contribution

It introduces novel local smoothing estimates in α-modulation spaces for fractional Schrödinger equations, with sharpness results and applications to nonlinear equations.

Findings

New local smoothing estimates derived using decoupling inequalities.

Sharpness of estimates shown through necessary conditions.

Applications to local well-posedness of fourth-order nonlinear Schrödinger equations.

Abstract

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.