Regularity and pointwise convergence for dispersive equations on Riemannian symmetric spaces of compact type

TL;DR

This paper establishes pointwise convergence of solutions to dispersive equations on compact Riemannian symmetric spaces, improving regularity thresholds under certain symmetry conditions and introducing a novel transference principle.

Contribution

It improves regularity thresholds for pointwise convergence on symmetric spaces and introduces a new transference principle applicable to various dispersive equations.

Findings

Pointwise convergence holds for Sobolev regularity  > 1/2 on rank 1 and 2 symmetric spaces.

For K-biinvariant initial data, the threshold improves to  > 1/3, failing below  < 1/4 for Schr43dinger.

A new transference principle is developed, applicable to multiple dispersive equations and potentially of independent interest.

Abstract

In this article, we first prove that for general dispersive equations on Riemannian symmetric spaces of compact type $\mathbb{X}=U/K$, of rank $1$ and $2$, the Sobolev regularity threshold $\alpha >1/2$ for the initial data, is sufficient to obtain pointwise convergence of the solution a.e. on $\mathbb{X}$. We next focus on $K$-biinvariant initial data for certain special cases of rank $1$, depending on geometric and topological considerations, and prove that the sufficiency of the regularity threshold can be improved down to $\alpha>1/3$, whereas the phenomenon fails for $\alpha<1/4$ for the Schr\"odinger equation. We also obtain the same results for other dispersive equations: the Boussinesq equation and the Beam equation, also known as the fourth order Wave equation, by a novel transference principle, which seems to be new even for the circle $\mathbb{T} \cong SO(2)$ and may be of independent interest. Our arguments involve harmonic analysis arising from the representation theory of compact semi-simple Lie groups and also number theory.