Uncertainty Principles for the Strichartz Fourier transform on the Heisenberg Group

TL;DR

This paper develops fundamental uncertainty principles for the Strichartz Fourier transform on the Heisenberg group, extending classical results to a new scalar-valued transform framework.

Contribution

It introduces and proves several uncertainty principles for the Strichartz Fourier transform, including analogues of Benedicks', Donoho-Stark, Price's local, and Beurling's theorems.

Findings

Proved an analogue of Benedicks' theorem for the Strichartz Fourier transform.

Established Nazarov's uncertainty inequality in this setting.

Extended the local uncertainty principle of Price to the Heisenberg group 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.