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In this paper, we combine the arguments of [L. Fanelli, J. Zhang and J. Zheng, Uniform resolvent estimates for Schrödinger operators in critical magnetic fields, Int. Math. Res. Not. IMRN 2023), 10.1093/imrn/rnac362] and [Y. Sire, C. D. Sogge, C. Wang and J. Zhang, Reversed Strichartz estimates for wave on non-trapping asymptotically hyperbolic manifolds and applications, Comm. Partial Differential Equations 47 2022, 6, 1124–1132] to prove the maximal estimates for fractional Schrödinger equations(i⁢∂t+ℒα2)⁢u=0{(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, forα=1{\alpha=1}, the maximal estimate for wave equation is sharp up to the endpoint.

Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields