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Studies of some Operators of Harmonic Analysis in certain Function Spaces with Applications to PDEs
Faculty of Science and Technology, Department of Computer Science and Computational Engineering, UiT The Arctic University of Norway, Narvik, Norway.ORCID iD: 0000-0002-6058-6512
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The study in this PhD thesis aims at development of certain mathematical methods used in applications, in particular, in the study of regularity properties of solutions in various mathematical models described by Partial Differential Equations (PDEs). To this end, we study various operators of harmonic analysis in certain function spaces, since solutions to many PDEs may be expressed in terms of such operators.

This PhD thesis consists of four papers (papers A–D) and an Introduction.

In Paper A we introduce a version of weighted anisotropic mixed norm Morrey spaces and anisotropic Hardy operators. We derive conditions for boundedness of these operators in such spaces. We also reveal the role of these operators in the solving of some degenerate hyperbolic PDEs of some class.

In Paper B we prove the boundedness of potential operators in weighted generalised Morrey space in terms of Matuszewska-Orlicz indices of weights and apply this result to the Helmholtz equationin R3 with a free term in such a space. We also give a short overview of some typical situations when potential type operators arise when solving PDEs.

In Paper C we study the boundedness of some multidimensional Hardy type operators in Hölder spaces and derive some new results of interest also in the theory of inequalities.

In Paper D we prove some differentiation formulas for weighted singular integrals, which we suppose to apply in our future studies concerning the solution of some integral equations of the first kind.

These new results are put into a more general frame in an Introduction, where also crucial parts of previous research by the candidate (e.g. published in two Licentiate theses) are briefly described. Note, in particular, that this PhD thesis may be regarded as a more theoretically based continuation of the Licentiate thesis in Wood Technology. This important link is carefully described in the Introduction.

Place, publisher, year, edition, pages
Narvik: UiT The Arctic University of Norway , 2018. , p. 93
Keywords [en]
Operators, Hardy Operators, Potential Operators, Singular Operators, Function Spaces, Morrey Spaces, H¨older Spaces, Partial Differential Equations, Inequalities, Hardy Type Inequalities, Weights, Applications
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-71990ISBN: 978-82-7823-212-5 (print)ISBN: 978-82-7823-213-2 (electronic)OAI: oai:DiVA.org:ltu-71990DiVA, id: diva2:1269610
Available from: 2018-12-13 Created: 2018-12-11 Last updated: 2019-02-18Bibliographically approved

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http://hdl.handle.net/10037/14696

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Lundberg, Staffan

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