Maximal estimates for fractional Schrödinger equations in scaling critical magnetic fields

Abstract

Abstract 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 + ℒ 𝐀 α 2 ) ⁢ 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, for α = 1 {\alpha=1} , the maximal estimate for wave equation is sharp up to the endpoint.