Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space

Abstract

We study the pointwise decay estimates for the Schr\"odinger and wave equations on a product cone $(X,g)$, 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 $h$. Under the assumption that the {conjugate radius} $\conR$ of $Y$ satisfies $\conR>\pi$, we prove the pointwise dispersive estimates for the Schr\"odinger and half-wave propagators in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come into play if $\conR>\pi$. A new finding is that a threshold of the {conjugate radius} of $Y$ for the pointwise dispersive estimates in this setting is the magical number $\pi$.