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

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 thresholds for the initial data, $\alpha >1/2$ and $\alpha >1$ respectively, are sufficient to obtain pointwise convergence of the solution a.e. on $\mathbb{X}$. We next focus on $K$-biinvariant initial data for rank $1$ 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.