Uncertainty principles for the Strichartz Fourier transform on the Heisenberg group

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 in 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 a quantitative analogue of Benedicks' theorem. 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.