Post-graduate theses
Current Record: 28 of 127
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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
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Κεραμίδας, Βασίλειος
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Thesis advisor
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Μήτσης Θεμιστοκλής
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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.
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Language |
Greek, English |
Subject |
Covering lemma for rectangles |
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Λήμμα κάλυψης στα ορθογώνια |
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Μεγιστικός τελεστής |
Issue date |
2022-07-22 |
Collection
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School/Department--School of Sciences and Engineering--Department of Mathematics and Applied Mathematics--Post-graduate theses
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Type of Work--Post-graduate theses
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Permanent Link |
https://elocus.lib.uoc.gr//dlib/5/a/8/metadata-dlib-1664357797-962527-19282.tkl
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Views |
330 |