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

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 \(supp (f)\).