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Some Versions of Beurling's Theorem on H-type Groups

Aparajita Dasgupta, Prerna Gulia, Sanjoy Pusti, Sundaram Thangavelu

TL;DR

The paper proves Beurling-type uncertainty principles for H-type groups by combining Radon-transform methods with a Gutzmer-formula framework tailored to the H-type motion group. It develops Radon-based reductions to the Heisenberg setting and establishes Gutzmer identities for center dimensions $m=2$ and $m=3$, including both the original and reduced H-type groups. The results hinge on class-one representations, Laguerre spectral data, and holomorphic extension of the group Fourier transform, yielding rigidity: suitably decaying functions with compatible spectral decay must vanish. Overall, the work extends Beurling-type theorems from the Heisenberg group to a broader class of nilpotent Lie groups with controlled automorphism structure, providing new tools for uncertainty principles on noncommutative groups. The combination of Radon-transform reductions and Gutzmer-type identities offers a robust route to sharp Beurling-type results in nonabelian harmonic analysis.

Abstract

We prove an analogue of Beurling's theorem on the H-type groups of certain dimensions after establishing the Gutzmer's formula for the H-type groups. We also obtain some other versions of the theorem using the modified Radon transform.

Some Versions of Beurling's Theorem on H-type Groups

TL;DR

The paper proves Beurling-type uncertainty principles for H-type groups by combining Radon-transform methods with a Gutzmer-formula framework tailored to the H-type motion group. It develops Radon-based reductions to the Heisenberg setting and establishes Gutzmer identities for center dimensions and , including both the original and reduced H-type groups. The results hinge on class-one representations, Laguerre spectral data, and holomorphic extension of the group Fourier transform, yielding rigidity: suitably decaying functions with compatible spectral decay must vanish. Overall, the work extends Beurling-type theorems from the Heisenberg group to a broader class of nilpotent Lie groups with controlled automorphism structure, providing new tools for uncertainty principles on noncommutative groups. The combination of Radon-transform reductions and Gutzmer-type identities offers a robust route to sharp Beurling-type results in nonabelian harmonic analysis.

Abstract

We prove an analogue of Beurling's theorem on the H-type groups of certain dimensions after establishing the Gutzmer's formula for the H-type groups. We also obtain some other versions of the theorem using the modified Radon transform.

Paper Structure

This paper contains 11 sections, 19 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Radon transform
  4. Proof of Theorem \ref{["first Beurling's using radon transform"]} and Theorem \ref{["second Beurling's using radon transform"]}
  5. Gutzmer's Formula
  6. The case when $m=3$:
  7. The case when $m=2$:
  8. Gutzmer's formula on reduced H-type group
  9. The case when $m=3$:
  10. The case when $m=2$:
  11. Proof of theorem \ref{["Beurling's theorem"]}

Key Result

Theorem 1.1

For $f \in L^2\left(\mathbb{H}^n\right)$, assume that $\mathcal{R}(\hat{f}(\lambda)) \subset \mathcal{D}\left(\pi_\lambda(z, w)\right)$ and $\pi_\lambda(z, w) \hat{f}(\lambda)$ is of trace class for all $(z, w) \in \mathbb{C}^{2 n}$ such that If the condition holds for almost every $\lambda \in \mathbb{R}^*$, then $f=0$.

Theorems & Definitions (32)

  • Theorem 1.1: Beurling's theorem on $\mathbb{H}^n$
  • Theorem 1.2: Modified Beurling's theorem on $\mathbb{H}^n$
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Lemma 2.2
  • Remark 3.1
  • proof : Proof of Theorem \ref{["first Beurling's using radon transform"]}:
  • proof : Proof of Theorem \ref{["second Beurling's using radon transform"]}:
  • ...and 22 more