Decay and Strichartz estimates for critical electromagnetic wave equations on conic manifolds

TL;DR

This paper proves decay and Strichartz estimates for wave equations with large, critical electromagnetic potentials on conic manifolds, advancing understanding of wave behavior in singular geometric settings.

Contribution

It introduces new localized spectral measure techniques to handle large critical electromagnetic potentials on conic manifolds, extending prior results to more general and larger potentials.

Findings

Established decay estimates for wave equations with critical potentials

Derived Strichartz estimates in conic geometries

Extended previous results to larger, scaling-critical potentials

Abstract

We establish the decay and Strichartz estimates for the wave equation with large scaling-critical electromagnetic potentials on a conical singular space $(X,g)$ with dimension $n\geq3$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. The decay assumption on the magnetic potentials is scaling critical and includes the decay of Coulomb type. The main technical innovation lies in proving localized pointwise estimates for the half-wave propagator by constructing a localized spectral measure, which effectively separates contributions from conjugate point pairs on $\CS$. In particular, when $Y=\mathbb{S}^{n-1}$, our results, which address the case of large critical electromagnetic potentials, extend and improve upon those in [21], which considered sufficiently decaying, and small potentials and that of [24], which considered potentials decaying faster than scaling critical ones.