Majorants of functions with zero integrals over balls of fixed radius
type: Item
instance of: scholarly article
title: Majorants of functions with zero integrals over balls of fixed radius
zbMATH Open document ID: 1162.53056
DOI: 10.1134/S1064562408030332
published in: Doklady Mathematics
publication date: 2008-11-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 44A35
zbMATH DE Number: 5363486
zbMATH Keywords: ball means
zbMATH Keywords: Liouville type theorems
zbMATH Keywords: the support problem
author: O. A. Ochakovskaya
MaRDI profile type: Publication
cites work: Q4437947
cites work: Q3228825
cites work: Spherical means and CR functions on the Heisenberg group
cites work: Liouville-type theorems for functions with zero integrals over balls of fixed radius
cites work: Q3892486
full work available at URL: s1064562408030332
OpenAlex ID: W2014633717
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description: scientific article; zbMATH DE number 5363486