Decay estimates for one Aharonov-Bohm solenoid in a uniform magnetic field III: Product cones

Abstract

The goal of a recently launched project is to extend the Euclidean models in \cite{Wang24,WZZ25-AHP,WZZ25-JDE} to a more general setting of conically singular spaces. In this paper, the main results include a weighted dispersive inequality for the Schr\"odinger equation and a dispersive estimate for the wave equation both with one Aharonov-Bohm solenoid in a uniform magnetic field on the product cone $X=\mathcal{C}(\mathbb{S}_\sigma^1)=(0,+\infty)_r\times\mathbb{S}_\sigma^1$ endowed with the flat metric $g=dr^2+r^2d\theta^2$, where $\mathbb{S}_\sigma^1\simeq\mathbb{R}/2\pi\sigma\mathbb{Z}$ denotes the circle of radius $\sigma\geq1$ in the Euclidean plane $\mathbb{R}^2$. As a byproduct, we also give the corresponding Strichartz estimates for these equations via the abstract argument of Keel-Tao.