Pseudodifferential operators on compact abelian groups with applications. - Sam Rumbelow (2006)
Pseudodifferential operators on compact groups are discussed, with an emphasis on the conditions for which the theorem of Hille and Yosida holds. Some preliminary functional analysis is given including the notion of regularly dissipative operators and Pontrjagin duality. The dual group is described, especially that it is discrete. Some important inequalities, such as Young's inequality, are also stated. Generalised trigonometrical polynomials and generalised Sobolev spaces are defined on the compact group G. A finite exhaustion of the dual space is used to define pointwise convergence and to give a condition for which a generalised Sobolev space is continuously embedded in C(G) and compactly embedded into a larger Sobolev space. The thesis defines k-ellipticity, k-smoothing operators and the k-parametrix, and proves their relation to the compactness of the embedding. It is shown that k-ellipticity is characterised by an inequality of Garding type. Some examples of pseudodifferential operators with constant coefficients are given. Another inequality of Garding type is proved for pseudodifferential operators with variable coefficients, and the existence of a weak solution to (A(x,D) - lambda)u = f is given under certain conditions on the adjoint A*(x,D). A variational solution of B[ϕ,u] = (ϕ,f) is found, and we prove a Garding type inequality for the sesquilinear form.