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dc.contributor.advisorWright, Jimen
dc.contributor.advisorBlue, Pieteren
dc.contributor.authorBakas, Odysseasen
dc.date.accessioned2018-03-13T14:59:40Z
dc.date.available2018-03-13T14:59:40Z
dc.date.issued2017-11-30
dc.identifier.urihttp://hdl.handle.net/1842/28797
dc.description.abstractThis thesis is motivated by the problem of understanding the endpoint mapping properties of higher-dimensional Marcinkiewicz multipliers. The one-dimensional case was definitively characterised by Tao and Wright. In particular, they proved that Marcinkiewicz multipliers acting on functions over the real line map the Hardy space H¹(ℝ) to L¹;∞(ℝ) and they locally map L log¹/² L to L¹;∞ and that these results are sharp. The classical inequalities of Paley and Zygmund involving lacunary sequences can be regarded as rudimentary prototypes of the aforementioned results of Tao and Wright on the behaviour of Marcinkiewicz multipliers "near" L¹(ℝ). Motivated by this fact, in Chapter 3 we obtain higher-dimensional variants of these two inequalities and we establish sharp multiplier inclusion theorems on the torus and on the real line. In Chapter 4 we extend the multiplier inclusion theorem on T of Chapter 3 to higher dimensions. In the last chapter of this thesis, we study endpoint mapping properties of the classical Littlewood-Paley square function which can essentially be regarded as a model Marcinkiewicz multiplier. More specifically, we give a new proof to a theorem due to Bourgain on the growth of the operator norm of the Littlewood- Paley square function as p → 1+ and then extend this result to higher dimensions. We also obtain sharp weak-type inequalities for the multi-parameter Littlewood- Paley square function and prove that the two-parameter Littlewood-Paley square function does not map the product Hardy space H¹ to L¹;∞.en
dc.language.isoen
dc.publisherThe University of Edinburghen
dc.relation.hasversionOdysseas Bakas. Endpoint mapping properties of the littlewood-paley square function. arXiv preprint arXiv:1612.09573, 2016.en
dc.relation.hasversionOdysseas Bakas. Variants of the Inequalities of Paley and Zygmund. arXiv preprint arXiv:1702.07049, 2017.en
dc.relation.hasversionOdysseas Bakas. A sharp multiplier inclusion theorem on product domains. arXiv preprint arXiv:1706.01712, 2017.en
dc.subjectMarcinkiewicz multipliersen
dc.subjectmappingen
dc.subjectLittlewood-Paley square functionen
dc.titleMapping properties of multi-parameter multipliersen
dc.typeThesis or Dissertationen
dc.type.qualificationlevelDoctoralen
dc.type.qualificationnamePhD Doctor of Philosophyen


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