On the injectivity of the Pompeiu transform for integral ball means (Q765390)

The Pompeiu transform \(f\to P_rf\) is defined as a mapping from \(L^1_{\text{loc}}(\mathbb{R}^n)\) by the formula \[ P_r f(x)= \int_{|y|\leq r} f(x+ y)\,dy. \] In this paper the author deals with the problem of infectivity of this transform. The main result describes conditions under which the equation \(P_r f(x)= 0\) for all \(x\in\mathbb{R}^n\) leads to \(f=0\).