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A great deal of classical harmonic analysis is concerned with the properties of the Laplacian L on Euclidean space Rn and on the sphere Sn−1 in Rn. One of the key questions which inspired many of the experts on the subject was the following. Given a bounded Borel function m on Rn, we can define a bounded operator m(L) on L2(Rn) using the functional calculus of self-adjoint operators on a Hilbert space. What conditions on m ensure that this operator extends to a bounded opera-tor on Lp(Rn) when p 6 = 2? Many of the answers to this question find their way into the theory of partial differential equations. There are versions of L. Hörmander’s classic multiplier theorem [7, Theorem 2.5] which hold for the Laplace–Beltrami operator on the sphere, due to R.R. Coifman and G. Weiss [3] and A. Bonami and J.-L. Clerc [2]. In this thesis, we are concerned with a related operator on the sphere. In complex analysis, one of the directions tangent to the sphere is atypical, namely that which is i times the radial direction. Problems in complex analysis about th