Smoothing effects of dispersive equations on real rank one symmetric spaces

TL;DR

This paper demonstrates that dispersive pseudodifferential equations exhibit global smoothing effects on real rank one symmetric spaces, linking decay rates of initial data to regularity gains, using advanced harmonic analysis tools.

Contribution

It extends smoothing effect results from Euclidean spaces to noncompact symmetric spaces using Helgason's Fourier transform and Radon transform techniques.

Findings

Proves time-global smoothing effects on symmetric spaces.

Analyzes regularity gains based on initial data decay.

Utilizes harmonic analysis tools like Fourier and Radon transforms.

Abstract

In this article we prove time-global smoothing effects of dispersive pseudodifferential equations with constant coefficient radially symmetric symbols on real rank one symmetric spaces of noncompact type. We also discuss gain of regularities according to decay rates of initial values for the Schroedinger evolution equation. We introduce some isometric operators and reduce the arguments to the well-known Euclidean case. In our proof, Helgason's Fourier transform and the Radon transform as an elliptic Fourier integral operator play crucial roles.