Almost-everywhere convergence of Bochner-Riesz means on Heisenberg-type groups

Horwich, Adam Daniel (2019). Almost-everywhere convergence of Bochner-Riesz means on Heisenberg-type groups. University of Birmingham. Ph.D.

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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.