Analysis of a Subelliptic Operator on the Sphere in Complex N -space

A great deal of classical harmonic analysis is concerned with the properties of the Laplacian L on Euclidean space R and on the sphere Sn−1 in R . One of the key questions which inspired many of the experts on the subject was the following. Given a bounded Borel function m on R , we can define a bounded operator m(L) on L(R) using the functional calculus of self-adjoint operators on a Hilbert space. What conditions on m ensure that this operator extends to a bounded operator on L(R) 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 the boundary behaviour of holomorphic functions are resolved using a modified version of the usual Laplacian, known as the Kohn–Laplacian, in which the square of the partial derivative in the ‘atypical’ direction is omitted from the usual Laplacian. We study this operator using the complex analogue of the theory of spherical harmonics (see [9] and [1] for the case of the sphere in R ). To do so, in this thesis, we quickly summarise a few properties of harmonic functions, and then recall the theory of spherical harmonics on the ‘real sphere’. We then develop an analogous theory for the ‘complex sphere’, which is certainly known to the experts, but we give a leisurely presentation which does not rely on previous exposure to the representation theory of the unitary group. Finally, we develop some ‘weighted estimates’ for spherical harmonics which enable us to prove a ‘weighted Plancherel theorem’ for the Kohn–Laplacian. This result, together with the general theorem of M. Cowling and A. Sikora [5], enable us to assert that an analogue of the Hörmander multiplier theorem holds for the Kohn–Laplacian, where the index is optimal.