Variations on a theorem of Cowling and Price with applications to nilpotent Lie groups

Abstract

In this paper we prove a new version of the Cowling-Price theorem for Fourier transforms on R". Using this we formulate and prove an uncertainty principle for operators. This leads to an analogue of the Cowling-Price theorem for nilpotent Lie groups. We also prove an exact analogue of the Cowling-Price theorem for the Heisenberg group.2000 Mathematics subject classification: primary 43A30.

The norm of the Fourier transform on compact or discrete abelian groups

“…We mention in passing that it is, of course, of great interest to obtain norms of Fourier transforms, and sharp constants in uncertainty principles, for nonabelian locally compact groups (starting with the most common Lie groups) or other general structures such as metric measure spaces, but this is a much harder project that we do not address at all. For the current state of knowledge on such questions, the reader may consult [6,26,41,8,32] and references therein.…”

Variants of Miyachi’s Theorem for Nilpotent Lie Groups

“…The Fourier transformq a (λ) of the heat kernel is the operator e −a H (λ) , where H (λ) = − + λ 2 |x| 2 is the scaled Hermite operator on R n . Thus the Hardy condition on the Fourier transform side reads f (λ)e a H (λ) op ≤ C. Generalizing Hardy's theorem, a version of the Cowling-Price theorem was obtained in [11]. Following similar ideas, we can prove the following version of Miyachi's theorem for the Heisenberg group.…”

Beurling's theorem on the Heisenberg group

“…By replacing g λ by L 2 functions and B λ by Hilbert-Schmidt operators, we get an analogue of Cowling-Price theorem on the Heisenberg group. For the general L p − L q version of Cowling-Price theorem, with a different proof, we refer to the work Parui-Thangavelu [7].…”