Maximal estimates for fractional Schr\"odinger equations in scaling critical magnetic fields

In this paper, we combine the argument of [12] and [27] to prove the maximal estimates for fractional Schr\"odinger 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.