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Bochner-Riesz Means on Heisenberg-Type Groups

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If you have a question about this talk, please contact Diogo Oliveira E Silva.

The question of the range of p for which Bochner-Riesz means of a function f in Lp converge to f has been well studied on R^n in an attempt to prove the Bochner-Riesz conjecture, which has been proven for n=2 but remains open for n>2. For p not smaller than 2, a similar result is expected for maximal Bochner-Riesz means, and this has led to questions of almost-everywhere convergence, a weaker result implied by Lp convergence of the maximal Bochner-Riesz means. The almost-everywhere result was proven by Carbery, Rubio de Francia and Vega (1988) and has been extended to Heisenberg groups by Gorges and Muller (2002), where Bochner-Riesz means are now defined in terms of the sub-Laplacian on these groups.

We prove result regarding convergence of Bochner-Riesz means on Heisenberg-type (H-type) groups, a class of 2-step nilpotent Lie groups that includes the Heisenberg groups. We broadly follow the method of Gorges and Muller, which in term is an adaption of techniques used by Carbery, Rubio de Francia and Vega. The implicit results in both papers, which reduce estimates for the maximal Bochner-Riesz operator from Lp to weighted L2 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. Obtaining these estimates requires an understanding of certain special functions (in particular Jacobi polynomials).

This talk is part of the Analysis Seminar series.

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