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A local version of the two-circles theorem. (English) Zbl 0624.31002

In [L. Zalcman, Arch. Ration. Mech. Anal. 47, 237-254 (1972; Zbl 0251.30047) and L. Brown, B. M. Schreiber and 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 F. John on the Euler-Darboux equation [Math. Ann. 111, 541-559 (1935; Zbl 0012.25402)], in fact, 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 \(T_ 0\).

MSC:

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
53C65 Integral geometry
45F05 Systems of nonsingular linear integral equations
42A75 Classical almost periodic functions, mean periodic functions
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