Characterization of Sobolev Spaces Based on Integral Averages
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Journal Title
Journal ISSN
Volume Title
Perustieteiden korkeakoulu |
Master's thesis
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Author
Date
2021-01-26
Department
Major/Subject
Mathematics
Mcode
SCI3054
Degree programme
Master’s Programme in Mathematics and Operations Research
Language
en
Pages
51+3
Series
Abstract
Characterizations of Sobolev spaces have attracted significant attention lately. A particularly promising characterization based on integral averages over balls was discovered recently that has inspired an abundance of new research, including an alternative approach for the original proof that is based on tools of harmonic analysis. This approach is stated to simplify the original proof and can potentially promote generalizing the characterization from the Euclidean setting to metric spaces. This thesis studies the alternative approach in detail. We restructure the proof to make it more tractable and present it explicitly and transparently. Furthermore, we analyze thoroughly the main tools of harmonic analysis that are used, namely the boundedness of the Littlewood-Paley operators and of the Marcinkiewicz integral transform. The main contribution of this thesis is to provide transparency and insight to the characterization, and thus pave the way for further studies about its generalization to metric spaces. Furthermore, we facilitate subsequent studies by gathering information about the tools used and details of the proof that are scattered across literature or completely omitted.Description
Supervisor
Kinnunen, JuhaThesis advisor
Kinnunen, JuhaKeywords
Sobolev spaces, Marcinkiewicz integral transform, Littlewood-Paley functions, Riesz transform, analysis
