Uniqueness of the group Fourier transform on certain nilpotent Lie groups

Abstract

In this article, we prove that if the group Fourier transform of certain integrable functions on the Heisenberg motion group (or step two nilpotent Lie groups) is of finite rank, then the function is identically zero. These results can be thought as an analogue to the Benedicks theorem that dealt with the uniqueness of the Fourier transform of integrable functions on the Euclidean spaces.

Benedicks-Amrein-Berthier theorem for the Heisenberg motion group and quaternion Heisenberg group

“…Thereafter, Vemuri [15] replaced the compact support condition on the set B by finite measure. In [5] authors consider B as a rectangle in R 2n while proving an analogous result on step two nilpotent Lie groups and a version of this result, with a strong assumption on rank, derived therein for the Heisenberg motion group. Later in the article [9], this result is extended to arbitrary set B of finite measure for general step two nilpotent Lie groups.…”

“…The following orthogonality relation is derived in [5]. A version of this result also appeared in [13].…”

Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

“…Thereafter, Vemuri [33] replaced the compact support condition on B by finite measure. In [7] authors considered B as a rectangle in R 2n to prove analogous results in step two nilpotent Lie group and the Heisenberg motion group. In this article, we prove the result on the general step two nilpotent Lie group when B is an arbitrary set of finite measure, using the Hilbert space theory, though specifying the appropriate set of projections in the setups of general step two nilpotent Lie groups was a major bottleneck and sorted out.…”