Benedicks’ theorem for the Heisenberg group

Abstract

Abstract. If an integrable function f on the Heisenberg group is supported on the set B × R where B ⊂ C n is compact and the group Fourier transform f (λ) is a finite rank operator for all λ ∈ R \ {0}, then f ≡ 0.

Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

“…For f ∈ L 1 (R n ), both the sets {x ∈ R n : f (x) = 0} and {ξ ∈ R n : f (ξ) = 0} cannot possess finite Lebesgue measure simultaneously, unless f = 0. Later, various analogues of this result has been investigated in different aspects including the Heisenberg group and the Euclidean motion group (see [17,21,24,32,33]).…”

“…In [17], Narayanan and Ratnakumar proved that if f ∈ L 1 (H n ) is supported on B ×R, where B is a compact subset of C n , and f (λ) is of finite rank for each λ, then f = 0. Thereafter, Vemuri [33] replaced the compact support condition on B by finite measure.…”

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

“…In [12], Narayanan and Ratnakumar proved that if f ∈ L 1 (H n ) is supported on B × R, where B is a compact subset of C n , and f (λ) has finite rank for each λ, then f = 0. Thereafter, Vemuri [15] replaced the compact support condition on the set B by finite measure.…”

Heisenberg uniqueness pairs for some algebraic curves in the plane

“…That is, support of a function f ∈ L 1 (R n ) and its Fourier transformf cannot be of finite measure simultaneously. Later, various analogues of the Benedicks theorem have been investigated in different set ups, including the Heisenberg group and Euclidean motion groups (see [12,15,18]). …”