Use this URL to cite or link to this record in EThOS: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.751990
Title: Pseudodifferential operators on compact abelian groups with applications
Author: Rumbelow, Sam
Awarding Body: University of Wales, Swansea
Current Institution: Swansea University
Date of Award: 2006
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Abstract:
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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.751990  DOI: Not available
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