Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

TL;DR

This paper establishes a sharp spectral multiplier theorem for a sub-Laplacian on quaternionic spheres, advancing harmonic analysis on complex geometric structures with higher corank horizontal spaces.

Contribution

It introduces a novel spectral multiplier theorem for quaternionic spheres, the first on compact sub-Riemannian manifolds with horizontal space of corank greater than one.

Findings

Proves a sharp $L^p$ spectral multiplier theorem for quaternionic spheres.

Develops an elementary derivation of quaternionic spherical harmonic decomposition.

Extends harmonic analysis techniques to new geometric settings.

Abstract

A sharp $L^p$ spectral multiplier theorem of Mihlin--H\"ormander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation.