Horwich, Adam Daniel (2019). Almost-everywhere convergence of Bochner-Riesz means on Heisenberg-type groups. University of Birmingham. Ph.D.
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Horwich2019PhD.pdf
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Abstract
In this thesis, we prove a result regarding almost-everywhere convergence of Bochner–Riesz means on Heisenberg-type (H-type) groups, a class of 2-step nilpotent Lie groups that includes the Heisenberg groups \(H_{m}\). We broadly follow the method developed by Gorges and Müller [24] for the case of Heisenberg groups, which in turn extends techniques used by Carbery, Rubio de Francia and Vega [8] to prove a result regarding Bochner–Riesz means on Euclidean spaces. The implicit results in both papers, which reduce estimates for the maximal Bochner–Riesz operator from \(L_{p}\) to weighted \(L_{2}\) spaces and from the maximal operator to the non-maximal operator, have been stated as stand-alone results, as well as simplified and extended to all stratified Lie groups. We also develop formulae for integral operators for fractional integration on the dual of H-type groups corresponding to pure first and second layer weights on the group, which are used to develop ‘trace lemma’ type inequalities for H-type groups. Estimates for Jacobi polynomials with one parameter fixed, which are relevant to the application of the second layer fractional integration formula, are also given.
Type of Work: | Thesis (Doctorates > Ph.D.) | |||||||||
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Award Type: | Doctorates > Ph.D. | |||||||||
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Licence: | All rights reserved | |||||||||
College/Faculty: | Colleges (2008 onwards) > College of Engineering & Physical Sciences | |||||||||
School or Department: | School of Mathematics | |||||||||
Funders: | Engineering and Physical Sciences Research Council | |||||||||
Subjects: | Q Science > QA Mathematics | |||||||||
URI: | http://etheses.bham.ac.uk/id/eprint/9276 |
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