We prove that the boundary of a bounded domain is a set of injectivity for the twisted spherical means on C n for a certain class of functions on C n. As a consequence we obtain results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform.

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Let normalΓ be the hyperbola {false(x,yfalse)∈double-struckR2:xy=1} and Λβ be the lattice‐cross defined by normalΛβ=false(double-struckZ×{0}false)∪false({0}×βdouble-struckZfalse) in R2, where β is a positive real. A result of Hedenmalm and Montes‐Rodríguez says that (Γ,normalΛβ) is a Heisenberg uniqueness pair if and only if β⩽1. In this paper, we show that for a rational perturbation of Λβ, namely normalΛβθ=false(double-struckZ+false{θfalse}false)×false{0false}∪false{0false}×βdouble-struckZ,where θ=1/p,forsomep∈N and β is a positive real, the pair (Γ,normalΛβθ) is a Heisenberg uniqueness pair if and only if β⩽p.

Let Z(Ann(r, R)) be the class of all continuous functions f on the annulus Ann(r, R) in C n with twisted spherical mean f × µ s (z) = 0, whenever z ∈ C n and s > 0 satisfy the condition that the sphere S s (z) ⊆ Ann(r, R) and ball B r (0) ⊆ B s (z). In this paper, we give a characterization for functions in Z(Ann(r, R)) in terms of their spherical harmonic coefficients. We also prove support theorems for the twisted spherical means in C n which improve some of the earlier results.

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