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

TL;DR

This paper establishes maximal estimates for fractional Schrödinger equations in magnetic fields, including Aharonov-Bohm fields, using spectral measure techniques, with sharp results for the wave equation case.

Contribution

It introduces new maximal estimates for fractional Schrödinger equations in magnetic fields, extending previous methods to include Aharonov-Bohm fields and sharpens wave equation estimates.

Findings

Maximal estimates proven for fractional Schrödinger equations in magnetic fields.

Results include sharp estimates for the wave equation when α=1.

Spectral measure estimates are key to the proofs.

Abstract

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.