Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres (Q2304337)

This paper is devoted to the proof of a theorem of Mihlin-Hörmander type for a distinguished sub-Laplacian on the unit quaternionic sphere \(\mathbb S\) in \(\mathbb H^n\). The problem is to obtain sufficient conditions on a bounded multiplier \(m\), defined on \((0, +\infty)\), to imply that the operator \(m(\mathcal L)\) is bounded on \(L^p\) for \(p\) in some interval around \(2\). In this kind of results, multipliers are assumed to be locally in some fractional Sobolev space \(H^s\), \(s > 0\), and it is important to understand how small the smoothness order \(s\) can be in order to imply \(L^p\) boundedness. In the main result of the paper, the authors prove that, under the condition \(s>(4n-1)/2\), \(m(\mathcal L)\) is of weak type \((1, 1)\) and bounded on \(L^p (\mathbb S)\) for all \(1< p<\infty\). The exponent \((4n-1)/2\) (equal to half the topological dimension) is optimal, that is, it cannot be replaced by any smaller number. The proof hinges on a detailed and particularly clear description of the quaternionic spherical harmonic decomposition, which in turn depend on some already known properties of complex spherical harmonic decomposition. This is the first multiplier result that applies to a sub-Laplacian on a compact sub-Riemannian manifold of corank greater than 1.