On Octonionic Harmonic Projection Operators

Harmonic projection operators map square integrable functions on a sphere onto subspaces of spherical harmonics. Let \(S^{dn-1}\) denote the unit sphere in \(\mathbb R^{dn}\) , with \(d\in \{1,2,4,8\}\) . In this paper, we start to study these projectors in the octonionic setting, that is, when \(d=8\) and \(n=2\) . We also formulate a conjecture about the norm of harmonic projection operators, considered as operators from \(L^p (S^{dn-1})\) onto \(L^2 (S^{dn-1})\) , for \(1\le p\le 2\) and \(d\in \{1,2,4,8\}\) .