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Majorant problems in harmonic analysis Rains, Michael Anthony

Abstract

In various questions of Harmonic analysis we encounter the problem of deriving a norm inequality between a pair of functions when we know a (point wise) inequality between the transforms of these functions. Such problems are known as majorant problems. In this thesis we consider two related problems. First, in Chapter two, we extend the known results on the upper majorant property on compact abelian groups to noncompact locally compact abelian groups. We show, using various test spaces and two notions of majorant, that a Lebesgue space has the upper majorant property exactly when its index is an even integer or infinity. Furthermore, if a Lebesgue space has the lower majorant property, then the Lebesgue space with conjugate index has the upper majorant property. In the final chapter we consider the second problem. Here-, we are concerned with deriving global integrability conditions from local integrability conditions for functions which have nonnegative transforms. Such a property holds only in Lebesgue spaces whose index is an even integer or infinity. For Lebesgue spaces whose index is not an even integer or infinity the proof of the failure of this property is based on the failure of the majorant property in these spaces.

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