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.

Reviewer: C.A.Berenstein (College Park)

##### MSC:

53C65 | Integral geometry |

43A85 | Harmonic analysis on homogeneous spaces |

26B15 | Integration of real functions of several variables: length, area, volume |