Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields

Abstract

In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schrödinger equations $(i{\partial }_t+{\mathcal{L}}_{\mathbf{A}}^{\frac{\alpha }{2}})u=0$ in the purely magnetic fields which includes the Aharonov-Bohm fields. The proof is based on the cluster spectral measure estimates. In particular $\alpha =1$, the maximal estimate for wave equation is sharp up to the endpoint.

10 pages