Complexifications, Pseudo-Differential Operators, and the Poisson Transform

In this thesis, we study pseudo-differential operators on a real-analytic manifold, which is either compact Riemannian, or a Lie group with a bi-invariant metric. Our aim is to obtain algebras of such operators, acting on real-analytic functions, but preserving a tube domain into which the functions extend holomorphically. The tube domain, contained in a complexification, is known as a ”Grauert tube”. We show that all the operators commuting with the Laplacian have this property, and in so doing, we make use of the Poisson transform introduced by Stenzel [56]. The transform is derived from a special case of a claim by Boutet de Monvel in [3], which was proved only recently by Stenzel [55] and Zelditch [65] in different ways. We demonstrate that the same would be true of many other real-analytic operators, if Boutet de Monvel’s claim holds in general, and briefly discuss approaches to it. Finally, in the setting of operators on a Lie group carrying a bi-invariant metric, without using the transform, we obtain a non-trivial algebra with this property.
This algebra is determined by a subspace of the global matrix-valued symbols, which was introduced by Ruzhansky, Turunen and Wirth [49].