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Pseudodifferential operators on compact abelian groups with applications. / Sam Rumbelow

Swansea University Author: Sam Rumbelow

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,...

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Published: 2006
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42386
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spelling 2018-08-02T16:24:29.0569807 v2 42386 2018-08-02 Pseudodifferential operators on compact abelian groups with applications. 118ff3a10e3d7ba84f552c4491175a82 NULL Sam Rumbelow Sam Rumbelow true true 2018-08-02 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[&phiv;,u] = (&phiv;,f) is found, and we prove a Garding type inequality for the sesquilinear form. E-Thesis Mathematics. 31 12 2006 2006-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-02T16:24:29.0569807 2018-08-02T16:24:29.0569807 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Sam Rumbelow NULL 1 0042386-02082018162450.pdf 10798094.pdf 2018-08-02T16:24:50.3370000 Output 1847442 application/pdf E-Thesis true 2018-08-02T16:24:50.3370000 false
title Pseudodifferential operators on compact abelian groups with applications.
spellingShingle Pseudodifferential operators on compact abelian groups with applications.
Sam Rumbelow
title_short Pseudodifferential operators on compact abelian groups with applications.
title_full Pseudodifferential operators on compact abelian groups with applications.
title_fullStr Pseudodifferential operators on compact abelian groups with applications.
title_full_unstemmed Pseudodifferential operators on compact abelian groups with applications.
title_sort Pseudodifferential operators on compact abelian groups with applications.
author_id_str_mv 118ff3a10e3d7ba84f552c4491175a82
author_id_fullname_str_mv 118ff3a10e3d7ba84f552c4491175a82_***_Sam Rumbelow
author Sam Rumbelow
author2 Sam Rumbelow
format E-Thesis
publishDate 2006
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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[&phiv;,u] = (&phiv;,f) is found, and we prove a Garding type inequality for the sesquilinear form.
published_date 2006-12-31T03:52:52Z
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score 11.012678

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