Injectivity classes of the Pompeiu transformation (Q362513)

The Pompeiu transformation \(P_r\) corresponding to integral ball means is the mapping from \(L_{\text{loc}}^1({\mathbb R}^n)\) into \(C({\mathbb R}^n)\) defined by the equality \[ (P_r f)(x)=\int_{|y|\leq r} f(x+y)\,dy, \] where \(f\in L_{\text{loc}}^1({\mathbb R}^n)\), \(x \in {\mathbb R}^n\), and \(r>0\) is fixed. In this paper, the author obtains new conditions for the injectivity of the Pompeiu transformation for integral ball means.