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In the first part of this paper we study the local and global solvability and the hypoellipticity of a family of left-invariant sublaplacians ${\mathcal{L}}_{\alpha }$ on the spheres $S^{2n+1}\simeq U\left(n+1\right)/U\left(n\right)$. In the second part, we introduce a larger family of left-invariant sublaplacians ${\mathcal{L}}_{\alpha ,\beta }$ on $S^3\simeq SU\left(2\right)$ and study the corresponding properties by means of a Lie group contraction to the Heisenberg group.

We study the action of elementary shift operators on spherical functions on ordered symmetric spaces ${ℳ}_{m,n}$ of Cayley type, where $m\in \mathbb{N}$ denotes the multiplicity of the short roots and $n\in \mathbb{N}$ the rank of the symmetric space. For $m$ even we apply this to prove a Paley-Wiener theorem for the spherical Laplace transform defined on ${ℳ}_{m,n}$ by a reduction to the rank 1 case. Finally we generalize our notions and results to $B{C}_n$ type root systems.