On the boundary of Fourier and complex analysis: the Pompeiu problem. (Q1607715)

Let \(X\) be a locally compact topological vector space, \(\mu\) be a nonnegative Radon measure on \(X\), \(\{\gamma_j\}_{j=1}^N\) a finite collection of compacts subsets of \(X\), and \(G\) a topological group acting on \(X\) and leaving the measure invariant. The \textit{Pompeiu transform} is the map \(P:C(X)\to (C(G))^N\), \((P_jf)(g)=\int_{g\gamma_j}f\, d\mu\). The family \(\{\gamma_j\}\) is said to have the \textit{Pompeiu property} if \(P\) is injective and the \textit{Pompeiu problem } is to decide as explicitly as possible whether the family \(\{\gamma_j\}\) has the Pompeiu property. The author begins by surveying various variants of the Pompeiu problem and shows how they lead to many interesting problems in Fourier analysis and complex analysis. The main part of the paper is on Pompeiu problems for functions on the Heisenberg group \({\mathbf H}^n\). There is for example a characterization of CR functions on \({\mathbf H}^n\). The author proves a Morera type theorem for CR functions on the unit sphere in \({\mathbb C}^n\). In the final section of the paper the heat kernel for \({\mathbf H}^n\) is used to study a Pompeiu problem in the \(L^{p,q}\) classes on \({\mathbf H}^n\).