Pompeiu problem for the Heisenberg ball (Q449574)

Let \(H^n\) be the Heisenberg group of dimension \(2n+1\), i.e. the set \(\mathbb{C}^n\times \mathbb{R}\) with the group structure NEWLINE\[NEWLINE (z,t)(w,s)=\Big(z+w,t+s+2\text{Im} \langle z,w\rangle_{\mathbb{C}}\Big). NEWLINE\]NEWLINE In the paper under review the author develops an approach which extends the analysis of the Pompeiu problem in \(H^n\) from subsets of \(\mathbb{C}^n\times \{0\}\) to several concepts of the Heisenberg ball, of codimension zero in \(H^n\). In particular, interesting analogs of the two-radii theorem for Heisenberg balls which have the same dimension as the ambient space \(H^n\) are established. The extra dimension leads to extra complexity in the functions defining the conditions for the radii. In addition, the different concepts of the Heisenberg ball lead to different forms for these arithmetic conditions defining the radii.