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Extremal problems on Pompeiu sets. (English. Russian original) Zbl 0957.53042
Sb. Math. 189, No. 7, 955-976 (1997); translation from Mat. Sb. 189, No. 7, 3-22 (1998).
A compact subset $$K$$ of $$\mathbb{R}^n$$, $$n\geq 2$$, is said to have the Pompeiu property if, whenever a function $$f\in C(\mathbb{R}^n)$$ averages to zero on every set $$\sigma(k)$$, $$\sigma\in M(n)$$, the group of rigid motions of $$\mathbb{R}^n$$, then $$f\equiv 0$$. Since no ball has the Pompeiu property, one considers the same question for pairs $$B_1,B_2$$ of closed balls of different radii $$r_1, r_2$$, respectively. It is known that there is a countable exceptional set $$P_n$$ such that the pair $$B_1,B_2$$ will have the Pompeiu property if and only if $$r_1/r_2 \notin P_n$$. We refer to [L. Zalcman, Am. Math. Monthly 87, 161-175 (1980; Zbl 0433.53048)] for an introduction to this kind of problem and to the bibliography compiled by L. Zalcman in [NATO ASI Ser., Ser. C 365 185-194 (1992; Zbl 0830.26005), updates available from the author]. Typically this type of problem can be reduced to a problem of harmonic analysis of ideals in the space of distributions of compact support in $$\mathbb{R}^n$$. This depends in an essential way on the fact that one is posing the problem in the whole space. In [J. Anal. Math. 52, 133-166 (1989; Zbl 0668.30037)], R. Gay and this reviewer showed that one could prove local versions of this theorem, that is, $$\mathbb{R}^n$$ replaced by a ball $$B$$. Clearly, there is an obvious minimum size condition on $$B$$, namely that the radius of $$r$$ of $$B$$ is sufficiently large so that $$K$$ or a translate of $$K$$, fits into $$B$$. It turns out for the case of two balls $$B_1,B_2$$ of respective radii $$r_1,r_2$$, one needs that $$r_1+r_2<r$$. There are similar theorems for other $$K$$. In the present paper the author gives sharp bounds for this type of size conditions.

MSC:
 53C65 Integral geometry 43A85 Harmonic analysis on homogeneous spaces 26B15 Integration of real functions of several variables: length, area, volume
Pompeiu property
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