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Bilinear Littlewood-Paley Square Functions and Singular Integrals
dc.contributor.advisor | Torres, Rodolfo | |
dc.contributor.author | Hart, Jarod Victor | |
dc.date.accessioned | 2013-09-30T20:08:34Z | |
dc.date.available | 2013-09-30T20:08:34Z | |
dc.date.issued | 2013-05-31 | |
dc.date.submitted | 2013 | |
dc.identifier.other | http://dissertations.umi.com/ku:12720 | |
dc.identifier.uri | http://hdl.handle.net/1808/12329 | |
dc.description.abstract | In this dissertation we further develop the bilinear theory of vector valued Calderoón-Zygmund operators, Littlewood-Paley square functions, and singu- lar integral operators. These areas of harmonic analysis are motivated by po- tential theory, boundary value problems in partial differential equations, har- monic and analytic extension problems in complex analysis, and many other classical problems in analysis. Multilinear operator theory addresses difficul- ties that arise from product type operations in harmonic analysis. We first introduce Banach valued Calderoón-Zygmund operators in a bilinear setting, and prove weak endpoint estimates and interpolation results for them. By viewing Littlewood-Paley square functions as Calderoón-Zygmund operators taking values in a particular Banach space, we are able to obtain bounds of the square functions on product Lebesgue spaces for a complete set of in- dices. We give an in depth analysis of Littlewood-Paley square functions, which includes estimates on some products of smooth function spaces as well as the estimates on product Lebesgue spaces that are needed to apply the vec- tor valued Calderoón-Zygmund results. Finally, we prove boundedness criteria for a certain class of bilinear singular integral operators on product Lebesgue spaces using Littlewood-Paley square function techniques. We provide a new proof of the bilinear T1 theorem that does not rely on the linear version of the result. We also prove a bilinear Tb theorem, a result missing in the theory so far. The Littlewood-Paley square function techniques developed in this work are a powerful tool has potential to solve problems in areas like oscillatory integral operator theory, multiparameter operator theory, Fourier restriction, and non-linear partial differential equations. | |
dc.format.extent | 178 pages | |
dc.language.iso | en | |
dc.publisher | University of Kansas | |
dc.rights | This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author. | |
dc.subject | Mathematics | |
dc.subject | Bilinear | |
dc.subject | Calder´on- Zygmund theory | |
dc.subject | Harmonic analysis | |
dc.subject | Littlewood-paley | |
dc.subject | Singular integral | |
dc.title | Bilinear Littlewood-Paley Square Functions and Singular Integrals | |
dc.type | Dissertation | |
dc.contributor.cmtemember | Torres, Rodolfo | |
dc.contributor.cmtemember | Gavosto, Estela | |
dc.contributor.cmtemember | Orr, James | |
dc.contributor.cmtemember | Shao, Shuanglin | |
dc.contributor.cmtemember | Stefenov, Atanas | |
dc.thesis.degreeDiscipline | Mathematics | |
dc.thesis.degreeLevel | Ph.D. | |
kusw.oastatus | na | |
kusw.oapolicy | This item does not meet KU Open Access policy criteria. | |
kusw.bibid | 8086080 | |
dc.rights.accessrights | openAccess |
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Mathematics Dissertations and Theses [179]