arXiv: Classical Analysis and ODEs | 2019
Almost-everywhere convergence of Bochner-Riesz means on Heisenberg-type groups
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 Muller [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.