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 $2$ respectively, 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 certain special cases of rank $1$, depending on geometric 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 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.
From: Utsav Dewan [view email]
[v1]
Wed, 10 Dec 2025 14:36:42 UTC (25 KB)
[v2]
Thu, 25 Jun 2026 15:12:52 UTC (25 KB)
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