Localisation of Spectral Sums corresponding to the sub-Laplacian on the Heisenberg Group

Abstract

In this article we study localisation of spectral sums $\{S_R\}_{R > 0}$ associated to the sub-Laplacian $\mathcal{L}$ on the Heisenberg Group $\mathbb{H}^d$ where $S_R f := \int_0^R dE_{\lambda }f$, with $\mathcal{L} = \int_0^{\infty} \lambda \, dE_{\lambda}$ being the spectral resolution of $\mathcal{L}.$ We prove that for any compactly supported function $f \in L^2(\mathbb{H}^d)$, and for any $\gamma < \frac{1}{2}$, $R^{\gamma} S_R f \to 0$ as $ R \to \infty$, almost everywhere off $supp (f)$.