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dc.contributor.authorScurry, Jamesen_US
dc.date.accessioned2013-06-15T02:42:19Z
dc.date.available2013-06-15T02:42:19Z
dc.date.issued2013-02-19en_US
dc.identifier.urihttp://hdl.handle.net/1853/47565
dc.description.abstractThis thesis studies several problems dealing with weighted inequalities and vector-valued operators. A weight is a nonnegative locally integrable function, and weighted inequalities refers to studying a given operator's continuity from one weighted Lebesgue space to another. The case where the underlying measure of both Lebesgue spaces is given by the same weight is known as a one weight inequality and the case where the weights are different is called a two weight inequality. These types of inequalities appear naturally in harmonic analysis from attempts to extend classical results to function spaces where the underlying measure is not necessarily Lebesgue measure. For most operators from harmonic analysis, Muckenhoupt weights represent the class of weights for which a one weight inequality holds. Chapters II and III study questions involving these weights. In particular, Chapter II focuses on determining the sharp dependence of a vector-valued singular integral operator's norm on a Muckenhoupt weight's characteristic; we determine that the vector-valued operator recovers the scalar dependence. Chapter III presents material from a joint work with M. Lacey. Specifically, in this chapter we estimate the weak-type norms of a simple class of vector-valued operators, but are unable to obtain a sharp result. The final two chapters consider two weight inequalities. Chapter IV characterizes the two weight inequality for a subset of the vector-valued operators considered in Chapter III. The final chapter presents examples to argue there is no relationship between the Hilbert transform and the Hardy-Littlewood maximal operator in the two weight setting; the material is taken from a joint work with M. Reguera.en_US
dc.publisherGeorgia Institute of Technologyen_US
dc.subjectHarmonic analysisen_US
dc.subject.lcshHarmonic analysis
dc.titleOne and two weight theory in harmonic analysisen_US
dc.typeDissertationen_US
dc.description.degreePhDen_US
dc.contributor.departmentMathematicsen_US
dc.description.advisorCommittee Chair: Wick, Brett; Committee Member: Iliev, Plamen; Committee Member: Lacey, Michael; Committee Member: Lubinsky, Doron; Committee Member: Mitkovski, Mishkoen_US


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