[Paper Review] Uniqueness of the group Fourier transform on certain nilpotent Lie groups

Arup Chattopadhyay, Deb Kumar Giri+1 more|arXiv (Cornell University)|Jul 13, 2016

Mathematical Analysis and Transform Methods20 references3 citations

TL;DR

This paper establishes a uniqueness result for the group Fourier transform on the Heisenberg motion group and step two nilpotent Lie groups: if the Fourier transform of an integrable function with finite support has finite rank, then the function must vanish identically. The result extends Benedicks-type uncertainty principles to non-abelian nilpotent groups using representation theory and spectral analysis of operators on $ L^2 $-spaces.

ABSTRACT

In this article, we prove that if the group Fourier transform of certain integrable functions on the Heisenberg motion group (or step two nilpotent Lie groups) is of finite rank, then the function is identically zero. These results can be thought as an analogue to the Benedicks theorem that dealt with the uniqueness of the Fourier transform of integrable functions on the Euclidean spaces.

Motivation & Objective

• To extend Benedicks-type uncertainty principles to non-abelian nilpotent Lie groups, particularly the Heisenberg motion group.
• To investigate when the group Fourier transform of a compactly supported or finitely supported integrable function on such groups must vanish identically.
• To characterize the conditions under which the Fourier transform has finite rank, linking this to vanishing of the original function.
• To explore the spectral structure of the Fourier transform in terms of finite-rank operators on $ L^2 $-spaces associated with representations.
• To generalize results from Euclidean and Heisenberg group settings to more general step two nilpotent Lie groups via harmonic analysis and metaplectic representations.

Proposed method

• Utilizes the Stone-von Neumann theorem to parameterize irreducible unitary representations $ \pi_\lambda $ of the Heisenberg group $ \mathbb{H}^n $ by $ \lambda \in \mathbb{R}^* $.
• Employs the metaplectic representation $ \mu_\lambda(k) $ to describe intertwining operators for $ K = U(n) $-actions on $ L^2(\mathbb{R}^n) $, enabling analysis on the Heisenberg motion group $ G = \mathbb{H}^n \ltimes U(n) $.
• Defines the group Fourier transform via $ \hat{f}(\omega) = W_\omega(f^\omega) $, where $ W_\omega $ is a twisted convolution operator on $ L^2(\eta_\omega) $, and relates it to the spectral properties of $ f^\omega $.
• Applies the spectral theorem to decompose $ W_\omega(\bar{\tau}) $ as a finite sum of rank-one projections, showing that finite rank implies $ f^\omega = 0 $.
• Uses Plancherel's theorem and properties of special Hermite functions $ \phi_{\alpha\beta}^\lambda $ to analyze $ L^2 $-norms and support conditions.
• Employs the operator $ T(\phi,\psi)(v) $ and its relation to $ W_\omega $ to derive integral identities that force $ \tau \equiv 0 $ when $ W_\omega(\bar{\tau}) $ has finite rank.

Experimental results

Research questions

• RQ1 Under what conditions does the group Fourier transform of a compactly supported or finitely supported integrable function on the Heisenberg motion group vanish identically?
• RQ2 Can the finite-rank property of the group Fourier transform be used to deduce uniqueness in the context of step two nilpotent Lie groups?
• RQ3 How does the spectral structure of the Fourier transform on $ G = \mathbb{H}^n \ltimes U(n) $ relate to the support and vanishing of the original function?
• RQ4 Is there a non-trivial integrable function on a step two nilpotent Lie group whose Fourier transform has finite rank?
• RQ5 What role does the metaplectic representation play in extending uncertainty principles from the Heisenberg group to the Heisenberg motion group?

Key findings

• If $ f \in L^1 \cap L^2(G) $ is supported on a set $ \Sigma \subset \mathfrak{b} $ and its group Fourier transform $ \hat{f}(\omega) = W_\omega(f^\omega) $ has finite rank, then $ f = 0 $.
• For $ f^\omega \in L^1 \cap L^2(\mathfrak{b}) $ with finite support, if $ W_\omega(f^\omega) $ has finite rank, then $ f^\omega = 0 $, implying $ f = 0 $.
• When $ W_\omega(f^\omega) $ has rank one, the analysis via $ W_\omega(\bar{\tau}) $ leads to $ \tau \equiv 0 $, hence $ f^\omega = 0 $, proving the vanishing result.
• The finite-rank condition on $ W_\omega(f^\omega) $ forces the kernel $ K_y(\xi) $ in the integral representation of $ \tau_y(x) $ to be finitely supported, leading to $ \tau_y \equiv 0 $ for all $ y $.
• The spectral decomposition of $ W_\omega(\bar{\tau}) $ as a finite sum of rank-one operators implies that $ \tau $ is a finite sum of $ T(h_j, h_j) $, and finiteness of support forces $ h_j $ to be zero.
• By Plancherel's theorem, $ \|f\|_{L^2}^2 = 0 $, hence $ f = 0 $, when $ W_\omega(f^\omega) $ has finite rank and $ f^\omega $ is finitely supported.