Inversion of the local Pompeiu transformation on a quaternionic hyperbolic space.
Let \(X\) be a homogeneous Riemannian manifold, \(G\) the group of isometries of \(X\), \(d\mu\) the element of Riemannian measure on \(X\), \(o\in X\) a fixed point, and \(d(x,y)\) the Riemannian distance on \(X\) (\(x, y\in X\)). Let \(B(o, r)=\{x\in X: d(x,o)<r\}\) be the disk with radius \(r\) and center \(o\). The (global) Pompeiu transform \(\mathcal P\) for disks \(B_j=B(o,r_j)\) of distinct radii \(r_j\), \(j=1,2\), maps \(C(X)\) to \(C(G)\times C(G)\) and is defined by the equality \[ ({\mathcal P}(f))(g_1,g_2)=\left(\int_{g_1B_1}f\,d\mu, \int_{g_2 B_2}f\,d\mu\right), \qquad g_1,g_2\in G. \tag{1} \] Similarly, for a disk \(B(o, R)\), \(R>\max(r_1, r_2)\), the local Pompeiu transform acts according to (1) from \(C(B(o, R))\) into the Cartesian product \(C(G_1)\times C(G_2)\), where \(G_j=\{g\in G: gB_j\subset B(o,R)\}\), \(j=1,2\). The following problem is called the Pompeiu problem: find whether or not \(\mathcal P\) is injective and if \(\mathcal P\) is injective, find the inverse mapping. In the Euclidean case \(X={\mathbb R}^n\) the local Pompeiu problem was studied by \textit{C. A. Berenstein}, \textit{R. Gay} and \textit{A. Yger} [J. Anal. Math. 54, 259--287 (1990; Zbl 0723.44002)] and by \textit{V. Volchkov} [Sb. Math. 186, No. 6, 783--802 (1995; Zbl 0860.43007)]. In the paper under review the authors consider the local Pompeiu problem in the case when \(X=H^n({\mathbb Q})\) is the quaternion hyperbolic space. The authors give sufficient conditions for the injectivity of the local Pompeiu transform and construct the inverse of this transform.
scientific article; zbMATH DE number 2074959