Uncertainty Principles for the Strichartz Fourier transform on the Heisenberg Group

TL;DR

This paper develops a comprehensive uncertainty-principle framework for the Strichartz Fourier transform on the Heisenberg group $\mathbb{H}^n$, a scalar-valued analogue of the classical operator-valued transform. It extends key Euclidean results to the noncommutative setting by proving analogues of Benedicks' theorem, the Donoho–Stark principle, Price's local uncertainty, and a Beurling-type theorem, with Nazarov-type corollaries. The authors employ projections $P_V$ and $P_W$, dilation properties, and Laguerre/Bessel spectral tools to obtain quantitative bounds in the Strichartz context, bypassing operator-valued complications. The results illuminate how simultaneous localization is restricted in $\mathbb{H}^n$ under the Strichartz transform and provide a foundation for further extensions to H-type groups and beyond. Overall, the work bridges Euclidean uncertainty principles and noncommutative harmonic analysis through a robust scalar transform framework.

Abstract

In this article, we establish several fundamental uncertainty principles for the Strichartz Fourier transform on the Heisenberg group, including Benedicks' theorem, the Donoho-Stark principle, the local uncertainty principle of Price, and a weak form of Beurling's theorem. The Strichartz Fourier transform, introduced by Thangavelu (2023), provides a scalar-valued analogue of the classical operator-valued Fourier transform on the Heisenberg group. We first prove an analogue of Benedicks' theorem asserting that a nonzero function and its Strichartz Fourier transform cannot both be supported on sets of finite measure. As a consequence, we obtain Nazarov's uncertainty inequality. We then establish the Donoho-Stark principle, providing quantitative bounds on simultaneous concentration in space and frequency, and extend the local uncertainty principle of Price to this framework. Finally, we present a weak form of Beurling's theorem for radial functions on the Heisenberg group.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1
• proof