Maximal Operators Defined by Rearrangement Invariant Banach Function Spaces

Bachelor thesis (2020)

Authors

D.M. Bonnema Electrical Engineering, Mathematics and Computer Science

Contributors

E. Lorist Analysis - EEMCS (mentor)
M.C. Veraar Analysis - EEMCS (graduation committee member)
Faculty
Electrical Engineering, Mathematics and Computer Science, Electrical Engineering, Mathematics and Computer Science
Copyright: © 2020 D.M. Bonnema
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Published Date

2020-7

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

In this thesis we study the boundedness of a generalization of the Hardy-Littlewood maximal operator, involving rearrangement invariant Banach function space and indices of the spaces.
We first consider a classical proof of boundedness of the Hardy-Littlewood maximal operator on rearrangement invariant Banach function spaces. After establishing necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, we consider a generalization of the Hardy-Littlewood maximal operator introduced by C. Pérez. We investigate and slightly improve the known sufficient conditions under which this more general maximal operator is bounded on a rearrangement invariant Banach function space. After which we search and find necessary conditions for boundedness in a general setting. In the final section we study Boyd indices and fundamental indices, especially how they are related to boundedness of the more general maximal operator. We also introduce weak fundamental indices and investigate some of their properties and uses. Finally we show how under certain assumptions we can state equivalent necessary and sufficient conditions for boundedness on Lorentz spaces and Orlicz spaces.

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