Localisation of Bochner Riesz means corresponding to the sub-Laplacian on the Heisenberg Group
Rahul Garg, Kaur Jotsaroop · arXiv (Cornell University) · 2018
In this article we prove localisation of Bochner Riesz means $S_R$ of order $0$ for the sub-Laplacian $\mathcal{L}$ on the Heisenberg Group $\mathbb{H}^d.$ More precisely, we show that for any $0 \eta$, $\lim_{R\rightarrow\infty}R^{-\beta/2} S_R f$ goes to $0$ a.e. on the set $\|(z,t)\|\leq 1$ for $f\in L^{2}(\mathbb{H}^d \setminus \{\|(z,t)\| \leq 3\}, \|(z,t)\|^{-\eta} \, dz \, dt).$ We generalise the method of Carbery and Soria (1988) in the context of $\mathbb{H}^d.$