Affiliation:
1. Department of Mathematics , Beijing Institute of Technology , Beijing 100081 , P. R. China
2. School of Science , China University of Geosciences (Beijing) , Beijing 100083 , P. R. China
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.
Subject
Applied Mathematics,General Mathematics