Two radii theorems on constant curvature spaces
Let \(f\in L^p_{\text{loc}}(R^n)\), \(n\geq 2\), be a function satisfying \(\int_{| u| \leq r}f(x+u)du=0\) for some fixed \(r>0\) and for all \(x\in R^n\). The so called Two Radii Theorem states that for \(f\) to vanish on \(R^n\) it is sufficient that the above integral condition holds only for two values \(r=r_1,r_2\), where \(r_1/r_2\) is different from the ratio of any two positive roots of the Bessel function \(J_{n/2}\). The author extends some of the known results (see, e.g., \textit{L. Zalcman} [Arch. Rat. Mech. Analysis 47, 237-254 (1972; Zbl 0251.30047)], \textit{C. A. Berenstein}, \textit{R. Gay} [Isr. J. Math. 55, 267-288 (1986; Zbl 0624.31002)], \textit{V. V. Volchkov} [Sb. Math. 186, 783-802 (1995; Zbl 0860.43007)]) considering two functions \(f_1\), \(f_2\) and asking for sufficient conditions for the equality \(f_1=f_2\). The sufficient conditions are given in terms of the growth rate of the integrals \(\int_{| u| \leq R}| f(u)| (1+| u|)^{(1-n)/2}du\) and \(\int_{| u| \leq R}| f(u)| ^pdu\), \(1\leq p\leq\frac{2n}{n-1}\), as \(R\to\infty\). Similar problems are considered also on \(n\)-dimensional hyperbolic spaces with constant curvature.
scientific article; zbMATH DE number 1113091