Let \(\mathbb{R}^n\) denote the Euclidean \(n\)-dimensional space for \(n\geq 2\), and let \(M(n)\) denote the motion group of \(\mathbb{R}^n\), which consists of functions of the form \(g: \mathbb{R}^n\to\mathbb{R}^n\) which preserve distances and orientations between points of \(\mathbb{R}^n\). The \(L^p\)-space \(L^p(E)\), \(E\subseteq\mathbb{R}^n\), with norm \(\|.\|_p\) is defined by \[ L^p(E)= \{[f], \|[f]\|_p=\| f\|_p<\infty\}, \] where \[ \| f\|_p= \Biggl(\int_E |f(x)|^p\,dx\Biggr)^{1/p},\;[f]= f\text{ a.e.} \] A non-empty open bounded set \(A\) in \(\mathbb{R}^n\) is said to have the Pompeiu property if \[ \int_{gA}f(x)\,dx= 0 \] for any locally integrable \(f\) and all \(g\in M(n)\) implies that \(f=0\). The subset \(A\) is said to have the weak Pompeiu property if \[ \int_{\mathbb{R}^n}|f(x)|(1+ |x|)^{-\alpha}\,dx< \infty, \] where \(\alpha> 0\), and \[ \int_{gA}f(x)\,dx= 0 \] for all \(g\in M(n)\) implies that \(f=0\). The characteristic function of a set \(A\) is denoted by \(\chi_A\), so that \(\chi_A(x)=1\) if \(x\in A\). If \(J_v\), \(v>-1\), denotes the Bessel function of the first kind with parameter \(v\), then the set \(E_{\lambda,n}\) is defined by \(E_{\lambda,n}= \{r> 0: J_{{1\over 2}n}(r\lambda)= 0\}\).
The statements of this paper indicate that a non-empty open set \(A\subseteq H\), the upper half plane, does not have the weak Pompeiu property but exhibits the Pompeiu property for a finite parameter \(p\geq 2\) if and only if there is \(\lambda= \lambda(A)> 0\) such that \(\chi_A\) is the limit of linear combinations of balls in \(H\) with radii in \(E_{\lambda,n}\).
The introduction of the paper notes that similar statements were indicated in the case \(p=1\) by V. V. Volchkov [Russ. Acad. Sci., Sb., Math. 79, No. 2, 281–286 (1994); translation from Mat. Sb. 184, No. 7, 71–78 (1993; Zbl 0821.35034)].