$L^p$-estimates for the wave equation with critical magnetic potential on conical manifolds

Abstract

In this paper, we consider a class of conical singular spaces $\Sigma =(0,\infty {)}_r\times Y$ equipped with the metric $g=\mathrm{d}{r}^2+r^{2}h$, where the cross section $Y$ is a compact $(n-1)$-dimensional closed Riemannian manifold $(Y,h)$ without boundary. In this context, we aim to show that the sine wave propagator $\sin \left(t\sqrt{{\mathcal{L}}_{\mathbf{A}}}\right)/\sqrt{{\mathcal{L}}_{\mathbf{A}}}$ is bounded in $L^p(\Sigma )$, where ${\mathcal{L}}_{\mathbf{A}}$ is a magnetic Schrödinger operator with a scaling-critical magnetic potential on metric cone $\Sigma$. Our main result is the generalization of the result in \cite{L}. The novel ingredient is the construction of Hadamard parametrix for $\cos \left(t\sqrt{{\mathcal{L}}_{\mathbf{A}}}\right)$ on $\Sigma$.