Analogues of theorems of Chernoff and Ingham on the Heisenberg group

TL;DR

This paper extends Chernoff's and Ingham's theorems to the setting of the Heisenberg group, providing new insights into the spectral analysis of the sublaplacian and Fourier transform on this non-commutative group.

Contribution

It introduces analogues of Chernoff's and Ingham's theorems for the Heisenberg group, advancing harmonic analysis in non-commutative settings.

Findings

Proved Chernoff's theorem analogue for the Heisenberg group's Laplacian.

Established Ingham type theorems for the group Fourier transform.

Derived spectral projection results related to the sublaplacian.

Abstract

We prove an analogue of Chernoff's theorem for the Laplacian $ \Delta_{\mathbb{H}} $ on the Heisenberg group $ \mathbb{H}^n.$ As an application, we prove Ingham type theorems for the group Fourier transform on $ \mathbb{H}^n $ and also for the spectral projections associated to the sublaplacian.