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Harmonic analysis on Chébli-Trimèche hypergroups

Xu, Zengfu (1994) Harmonic analysis on Chébli-Trimèche hypergroups. PhD thesis, Murdoch University.

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In this thesis we develop the theory on Chebli-Trimeche hypergroups of such topics as maximal functions, the convergence and boundedness of certain convolution operator families in Lp spaces and Hardy spaces as well as Fourier multipliers. As the basis of the theory we first investigate the Schwartz classes, Plancherel measure and hypergroup characters on these hypergroups, and establish basic facts about approximations to the identity and the important results concerning Fourier transforms and the estimates for the Plancherel measure and characters. These lead to estimates for the translation operator as well as the heat and Poisson kernels, all of which play a significant role in our study of various maximal operators. The latter include the Hardy-Littlewood maximal operator, the heat and Poisson maximal operators, a class of radial maximal operators, and the grand maximal operator. The behaviour of these maximal convolution operators on Lp and Hardy spaces is investigated, and some classical results are extended to Chebli-Trimeche hypergroups. We also develop local Hardy space theory, and give some results concerning Fourier multipliers and Riesz potentials.

Item Type: Thesis (PhD)
Murdoch Affiliation(s): School of Mathematical and Physical Sciences
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Supervisor(s): Bloom, Walter
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