Injectivity of the Pompeiu transform in the Heisenberg group (Q1326644)

Let \(X\) be a locally compact space, and \(G\) be a group acting on \(X\) and preserving a measure \(m\). If \(K\) is a compact set in \(X\) one defines the map \({\mathcal P}_ K : {\mathcal C} (X) \to {\mathcal C}(G)\) by \({\mathcal P}_ K (f)(g) = \int_{g(K)} fdm\). The Pompeiu problem is to decide whether \({\mathcal P}_ K\) is injective for a given \(K\). The map \({\mathcal P}_ K\) can be seen as a convolution operator, and Wiener Tauberian theorems can be used to solve the Pompeiu problem. In this paper \(X = \mathbb{H}_ n \simeq \mathbb{C}^ n \times \mathbb{R}\), the Heisenberg group of dimension \(2n + 1\), and one considers compact sets \(K\) which are \(\mathbb{T}^ n\)- or \(\mathbb{U}(n)\)-invariant. The convolution algebra \(L^ 1_ 0 (\mathbb{H}_ n)\) (resp. \(L^ 1_ *(\mathbb{H}_ n))\) of \(\mathbb{T}^ n\)- (resp. \(\mathbb{U}(n)\)-) invariant integrable functions is commutative. In the first part of the paper the authors determine the spectra of both algebras. (These results are not new. The spectrum of \(L^ 1_ * (\mathbb{H}_ n)\) has been determined by \textit{A. Korányi} [see Some applications of Gelfand pairs in classical analysis, in Harmonic analysis and group representations, CIME (1980)]). By using the corresponding Gelfand transform \(f \mapsto\widetilde{f}\) the authors prove the following result. Let \(\mathcal R\) be a family of \(\mathbb{T}^ n\)- (or \(\mathbb{U}(n)\)-) invariant and compactly supported measures. Assume that the functions \(\widetilde {T}\), for \(T \in {\mathcal R}\), have no common zero. Let \(f\) be a bounded continuous function such that \(f * T = 0\), \(\forall T \in \mathbb{R}\), then \(f \equiv 0\). This result is applied to the Pompeiu problem for a family of balls and a family of polydiscs.