Injectivity sets for spherical means on the Heisenberg group (Q6205148)

Let \( \mu _{r}\), \(r>0\), be the normalized surface measure on the sphere \(\{w\in {\mathbb C^n} : |w|=r\}\) in \({\mathbb C^n}\). For \(f\in C({\mathbb C^n})\), the twisted spherical means of \(f\) are defined as follows: \[ f\times \mu _r(z)=\int_{|w|=r} f(z-w)e^{i {1\over 2} \Im (z\cdot \overline w)} d\mu_r(w),\quad z\in {\mathbb C}^n. \] A set \(S\subset {\mathbb C}^n\) is said to be a set of injectivity for the twisted spherical means in a subclass \(\mathcal C\) of the continuous functions on \({\mathbb C}^n\) if \(f\in \mathcal C\) and \(f\times \mu_r(z)=0\), for all \(r\geq 0\) and \(z\in S\), imply that \(f=0\). The authors consider functions that satisfy \(f(z)e^{({1\over {4}}+\varepsilon)|z|^2}\in L^p({\mathbb C}^n)\) for some \(\varepsilon >0\) and \(1\leq p \leq \infty \). They show that in this class of functions, the boundary of a bounded domain in \({\mathbb C}^n\) is a set of injectivity for the twisted spherical means. As a consequence some results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform are given.