Majorants of functions with zero integrals over balls of fixed radius (Q954214)

Let \(\Omega\) be an unbounded domain in \(\mathbb{R}^n\), \(n\geq 2\), and let \(r>0\) be a fixed number. Denote by \(V_r(\Omega)\) the set of all functions \(f\in L_{loc}^{1}(\Omega)\) having zero integrals over all closed balls of radius \(r\) contained in \(\Omega\). The paper under review deals with the following problem. Problem 1. Suppose that \(f\in V_r(\Omega)\) and, for almost all \(x\in \Omega\), \[ |f(x)|\leq F(|x|), \] where \(F\) is a given positive function on \([0,+\infty)\), and \(|\cdot|\) is the Euclidean norm. For what \(F\), \(\Omega\), and \(r\) can we assert that \(f=0\)? All existing results on Problem 1 were concerned with the case where \(\Omega\) contains the exterior of a ball [see \textit{V. V. Volchkov}, Integral Geometry and Convolution Equations. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1043.53003)] (Parts 2 and 3, where a more general situation was considered in which the class \(V_r(\Omega)\) was replaced by the solution space of a convolution equation of special form). In the paper under review Problem 1 is solved for a large class of domains containing the half-space.