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 threshold $α>1/2$ 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 $α>1/3$, whereas the phenomenon fails for $α<1/4$ for the Schrödinger 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.