Injectivity sets of the Pompeiu transformation on the Euclidean spaces (Q2880022)

Some negative and positive results on the injectivity of the Pompeiu transformation are obtained. Suppose that \(M(n)\) is a group of movings in \({\mathbb R}^n,\) let \({\mathcal E}^{\,\prime}({\mathbb R}^n)\) be a space of distributions in \({\mathbb R}^n\) having a compact support. For \(\varphi\in {\mathcal E^{\,\prime}}({\mathbb R}^n)\) and \(\zeta\in M(n),\) define a distribution \(\zeta\varphi\) by the rule \(\left\langle\zeta\varphi, f(x)\right\rangle=\left\langle\varphi, f(\zeta^{-1}x)\right\rangle,\) where \(f\in {\mathcal E}^{\,\prime}({\mathbb R}^n).\) For an open set \({\mathcal U}\subset {\mathbb R}^n\) we consider a set \(\Lambda({\mathcal U}, \varphi)=\{\zeta\in M(n): \text{ supp\,}\zeta\varphi\subset {\mathcal U}\}\neq \varnothing.\) Theorem 1 of the work states the following. Suppose that \(\varphi\) is a distribution on the sphere \({\mathbb S}^{n-1},\) i.e., \(\varphi\in {\mathcal D}^{\,\prime}({\mathbb S}^{n-1}),\) and the spectrum of the distribution \(\varphi\) does not coincide with \(\{(0, 1)\},\) i.e., \(\text{ spec}\,\varphi\neq \{(0, 1)\}.\) Then for every \(\alpha>0\) the union of all of the open balls \({\mathcal U}\) contained in the ellipsoid \(\left\{x\in {\mathbb R}^n: (\alpha^2-1)(x_1^2+\cdots+x^2_{n-1})+\alpha^2x_n^2\leq \alpha^4\right\}\) satisfies the necessary condition of the injectivity of the Pompeiu transform, namely, \(\overline{\mathcal U}=\overline{\bigcup\limits_{\zeta\in\Lambda({\mathcal U}, \varphi)}\text{ supp\,}\zeta\varphi};\) however, the set \({\mathcal U}\) mentioned above does not belong to the set of injectivity of the Pompeiu transform. Moreover, for every open set \(E\subset {\mathbb R}^n\) containing two opposite points of the sphere it can be found the distribution \(\varphi\in {\mathcal D}^{\,\prime}({\mathbb S}^{n-1})\) for which \(\{(0, 1)\}\in \text{ spec\,}\varphi,\) \(\text{ supp}\,\varphi\subset E\cap{\mathbb S}^{n-1}\) and the condition of the type \(\overline{\mathcal E}=\overline{\bigcup\limits_{\zeta\in\Lambda({\mathcal E}, \varphi)}\text{ supp\,}\zeta\varphi}\) is not a sufficient condition for the injectivity of the Pompeiu transform. Finally, some additional requirements on the domain \({\mathcal U},\) to be an injectivity set, are given by the Theorems 3 and 4.