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

TL;DR

This paper investigates the localization properties of spectral sums related to the sub-Laplacian on the Heisenberg Group, showing decay of these sums outside the support of functions for certain growth rates.

Contribution

It establishes almost everywhere convergence and decay rates of spectral sums associated with the sub-Laplacian on the Heisenberg Group for compactly supported functions.

Findings

Spectral sums decay outside the support of functions.

Almost everywhere convergence of spectral sums.

Decay rate of $R^{eta} S_R f$ for $eta < 1/2$.

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 $supp (f)$.