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Identifier

000451337

Title

The LlogL inequality for the strong maximal function

Alternative Title

Η LlogL ανισότητα για την μεγιστική συνάρτηση

Author

Κεραμίδας, Βασίλειος

Thesis advisor

Μήτσης Θεμιστοκλής

Abstract

Throughout this thesis, we work on Rd with the d-dimensional Lebesgue measure. We investigate extensively the relation between the differentiation properties of a differentiation basis, the covering properties of a differentiation basis and the boundedness properties of the associated maximal operator. Some general results are presented, emphasising on necessary and sufficient geometric - covering conditions, under which a basis differentiates a space of functions. Furthermore, we highlight the significant geometric difference between balls and rectangles. That is, the volume of a rectangle may be arbitrarily small, while its diameter is arbitrarily big, whereas the volume of a ball is comparable with its diameter. Then, studying some covering lemmas of balls and cubes, we make a brief reference to the Hardy - Littlewood maximal operator. Finally, we focus on the basis of rectangles (with sides parallel to the coordinate axes) and the strong maximal operator, studying its differentiation properties as they are deduced by the geometry of rectangles. More specifically, we give some proofs of the so called LlogL inequality. We start with a technical proof, resulting from the boundedness properties of the HL maximal operator. The purpose of this thesis though, is to present the geometric proof of A. Cordoba and R. Fefferman. This proof constitutes a direct consequence of a suitable covering lemma for rectangles, involving no reference to the H-L maximal operator. We give two detailed relative proofs, firstly on R2 and then on Rd, d ≥ 2. In the final chapter, we shortly examine whether this basis differentiates any space worse than LlogL or not.

Language

Greek, English

Subject

Covering lemma for rectangles

Λήμμα κάλυψης στα ορθογώνια

Μεγιστικός τελεστής

Issue date

2022-07-22

Collection

School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses

Type of Work--Post-graduate theses

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