A local version of the two-circles theorem
In [\textit{L. Zalcman}, Arch. Ration. Mech. Anal. 47, 237-254 (1972; Zbl 0251.30047) and \textit{L. Brown, B. M. Schreiber} and \textit{B. A. Taylor}, Ann. Inst. Fourier 23, No. 3, 125-154 (1973; Zbl 0265.46044)] it is proven that if \(r_ 1,r_ 2>0\), \(r_ 1/r_ 2\not\in E_ n\), \(E_ n=\) set of quotients of positive zeros of the Bessel function \(J_{n/2}\), then the only function \(f\in C({\mathbb{R}}^ n)\) satisfying \(A_ j(x)=\int_{| y| >r_ j} f(x+y) dy=0\) for all \(x\in {\mathbb{R}}^ n\), \(j=1,2\), is the function \(f\equiv 0\). It turns out that this result can be obtained from the work of \textit{F. John} on the Euler-Darboux equation [Math. Ann. 111, 541-559 (1935; Zbl 0012.25402)], in fact, \textit{J. D. Smith} [Math. Proc. Camb. Philos. Soc. 72, 403-416 (1972; Zbl 0247.42021)] did obtain this way that if \(f\in C(| x| <R)\), \(r_ 1+2r_ 2<R\) and \(A_ j(x)=0\) for \(| x| <R-r_ j\) \((j=1,2)\) (i.e., as long as the ball \(\{| x-y| <r_ j\}\leq \{| y| <R\})\), then \(f\equiv 0\). By a different method we obtain the sharp form of this `local-two-circle' theorem, namely one only needs \(r_ 1+r_ 2<R\), furthermore, this condition is necessary. Our method does extend to other Pompeiu type problems. For instance, in this paper we obtain local versions of the work of Delsarte-Lions on harmonic functions. Let \(H_ n\) be the set of positive numbers that are quotients of complex zeros of the function \(2^{n/2-1}(\Gamma (n/2)J_{n/2-1}(z)/z^{n/2-1}.\) It is known that {\#}H\({}_ n<\infty\) and \(1\in H_ n\). Let \(r_ 1,r_ 2>0\) be such that \(r_ 1/r_ 2\not\in H_ n\) and \(r_ 1+r_ 2<R\). A function \(f\in C(| x| <R)\) is harmonic if and only if for \(j=1,2\) \[ \int_{| y| =1} f(x+r_ jy) d\sigma (y)=f(x)\quad whenever\quad | x| +r_ j<R \] (d\(\sigma\) \(=\) normalized measure in the unit sphere of \({\mathbb{R}}^ n).\) In a forthcoming paper we extend this type of theorems to the case where the balls of radius \(r_ 1\), \(r_ 2\) are replaced by sets congruent to a fixed set \(\Omega\). For instance, we can prove the following version of Morera's theorem: Let \(T_ 0\) be a fixed triangle \(\subseteq\{\) \(z\in {\mathbb{C}}:| z| <1/2\}\). A function \(f\in C(| z| <1)\) is holomorphic if and only if \(\int_{\partial T} f(z)dz=0\) for every triangle \(T\subseteq \{| z| <1\}\), congruent to \(T_ 0\).
scientific article; zbMATH DE number 4013008
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