A brief survey on muckenhoupt weights and Lp-boundedness of certain translation-invariant operators

Abstract

In this survey, we try to understand boundedness of translation-invariant (linear) operators on Lebesgue spaces. Translation-invariant operators are an important part of Fourier Analysis. These operators enjoy “nice”-properties. For instance, it is known, due to H¨ormander, that translation-invariant operators are “Lp-improving”. Our main aim is to see the boundedness of such operators on Lp-spaces of the Euclidean space not only with the usual Lebesgue measure, but also with measures induced by positive (measurable) functions. Such functions are referred to as weights. A natural question arises: Are all weights “good”? At the first glance, this question is ambiguous and does not merit an answer at all! How does one define “good” weights? A part of this thesis also describes some literature in this direction. Study of averages of functions on the real line was done nearly a century ago by Hardy and Littlewood, in the context of differentiability properties of integrable functions.