Smoothing properties of dispersive equations on non-compact symmetric spaces

TL;DR

This paper proves new smoothing estimates for dispersive equations on non-compact symmetric spaces, including hyperbolic spaces, by establishing key inequalities and extending comparison principles.

Contribution

It introduces the first Kato-type smoothing estimates for Schrödinger equations on non-compact symmetric spaces and extends these results to other dispersive equations.

Findings

Smoothing estimates hold on hyperbolic spaces, unlike Euclidean cases.

Stein-Weiss inequality and resolvent estimates are established for these spaces.

Comparison principles are extended to various dispersive equations.

Abstract

We establish the Kato-type smoothing property, i.e., global-in-time smoothing estimates with homogeneous weights, for the Schr\"odinger equation on Riemannian symmetric spaces of non-compact type and general rank. These form a rich class of manifolds with nonpositive sectional curvature and exponential volume growth at infinity, e.g., hyperbolic spaces. We achieve it by proving the Stein-Weiss inequality and the resolvent estimate of the corresponding Fourier multiplier, which are of independent interest. Moreover, we extend the comparison principles to symmetric spaces and deduce different types of smoothing properties for the wave equation, the Klein-Gordon equation, the relativistic and general orders Schr\"odinger equations. In particular, we observe that some smoothing properties, which are known to fail on the Euclidean plane, hold on the hyperbolic plane.