Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space
Qiuye Jia, Junyong Zhang
TL;DR
This work establishes sharp pointwise dispersive estimates for the Schrödinger and half-wave propagators on product cones with an inverse-square potential, under the assumption that the conjugate radius of Y exceeds π. The authors combine a spectral decomposition on Y with a modified Hadamard parametrix that remains valid up to time π, leveraging a distance-spectrum 𝔇 derived from geodesic data to build oscillatory integral parametrices. They prove L^p→L^{p'} dispersive bounds and, via Littlewood–Paley theory, derive Besov-based estimates and Strichartz-type results; a threshold π for L^p estimates emerges as a key geometric-dispersive indicator. The analysis unifies radial- angular separation, Bessel function asymptotics, FIO parametrices, and precise control of boundary terms, extending known results on cones and connecting dispersive decay to the geometry of conjugate points. The results have implications for Schrödinger and wave equations on singular spaces and highlight a clear dichotomy tied to the conjugate radius of Y.
Abstract
We study the pointwise decay estimates for the Schrödinger 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 $ε$ of $Y$ satisfies $ε>π$, we prove the pointwise dispersive estimates for the Schrödinger and half-wave propagator in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come to play if $ε>π$. In a work in progress, we will further study the case that $ε\leqπ$ in which the role of conjugate points come. A new finding is that a threshold of the conjugate radius of $Y$ for $L^p$-estimates in this setting is the magical number $π$.