Title:
One and two weight theory in harmonic analysis

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dc.contributor.advisor Wick, Brett D.
dc.contributor.author Scurry, James en_US
dc.contributor.committeeMember Iliev, Plamen
dc.contributor.committeeMember Lacey, Michael
dc.contributor.committeeMember Lubinsky, Doron
dc.contributor.committeeMember Mitkovski, Mishko
dc.contributor.department Mathematics en_US
dc.date.accessioned 2013-06-15T02:42:19Z
dc.date.available 2013-06-15T02:42:19Z
dc.date.issued 2013-02-19 en_US
dc.description.abstract This 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.description.degree PhD en_US
dc.identifier.uri http://hdl.handle.net/1853/47565
dc.publisher Georgia Institute of Technology en_US
dc.subject Harmonic analysis en_US
dc.subject.lcsh Harmonic analysis
dc.title One and two weight theory in harmonic analysis en_US
dc.type Text
dc.type.genre Dissertation
dspace.entity.type Publication
local.contributor.corporatename College of Sciences
local.contributor.corporatename School of Mathematics
relation.isOrgUnitOfPublication 85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication 84e5d930-8c17-4e24-96cc-63f5ab63da69
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