Analogues of theorems of Chernoff and Ingham on the Heisenberg group

Abstract

We prove an analogue of Chernoff's theorem for the Laplacian ∆ H on the Heisenberg group H n . As an application, we prove Ingham type theorems for the group Fourier transform on H n and also for the spectral projections associated to the sublaplacian.

An analogue of Ingham’s theorem on the Heisenberg group

“…Recently this aspect of Ingham's theorem has been proved in the context of higher dimensional Euclidean spaces and Riemannian symmetric spaces with a much weaker hypothesis on the function. As observed in [11], for the Heisenberg group case, the hypothesis can be weakened considerably if we slightly strengthen the condition (1.3). More precisely, the second and the last author proved the following theorem in this context.…”

“…These works mainly dealt with the Fourier transform on Riemannian symmetric spaces. The second and the last authors treated the case of Fourier transform on Heisenberg groups and eigenfunction expansions in [10] and [11]. In order to help the reader to get a better understanding of the status of the investigations in this interesting area of research, we would like to conclude this introduction with the following remarks.…”

“…Let us mention in passing that this new construction has already been used without proof in the work of [11]. More precisely, the authors in [11] convolved the function constructed in this paper with a suitable compactly supported function in the central variable in order to construct a function satisfying (1.4). See [11,Sect.…”

“…More precisely, the second and the last author proved the following theorem in this context. Theorem 1.4 [11] Let (λ) be a nonnegative function on [0, ∞) such that it decreases to zero as λ → ∞, and satisfies the conditions integrable function on H n whose Fourier transform satisfies the estimate…”

An analogue of Ingham's theorem on the Heisenberg group

“…Theorem 1.4. [11] Let Θ(λ) be a nonnegative function on [0, ∞) such that it decreases to zero as λ → ∞, and satisfies the conditions ∞ 1 Θ(t)t −1 dt = ∞. Let f be an integrable function on H n whose Fourier transform satisfies the estimate…”