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

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 Schrödinger equation and a dispersive estimate for the wave equation both with one Aharonov-Bohm solenoid in a uniform magnetic field on the product cone $X=\mathcal{C}({\mathbb{S}}_{\sigma }^1)=(0,+\mathrm{\infty }{)}_r\times {\mathbb{S}}_{\sigma }^1$ endowed with the flat metric $g=d{r}^2+r^{2}d{\theta }^2$, where ${\mathbb{S}}_{\sigma }^1\simeq \mathbb{R}\mathrm{/}2\pi \sigma \mathbb{Z}$ denotes the circle of radius $\sigma \ge 1$ 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.