zbMATH — the first resource for mathematics

On functions with zero integrals over ellipsoids. (English. Russian original) Zbl 1055.31004
Dokl. Math. 63, No. 1, 25-27 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 376, No. 2, 158-160 (2001).
A compact set \(A\subseteq\mathbb{R}^n\) is said to be a Pompeiu set on the ball \(B_r=\{x\in\mathbb{R}^n:| x|<r\}\) if every locally summable function \(f:B_r\to\mathbb{R}\) such that \(\int_{\lambda A}f(x)\,dx=0\) for all motions \(\lambda\) of \(\mathbb{R}^n\) for which \(\lambda A\subset B_r\) vanishes almost everywhere. For many compact sets \(A\), this holds if \(r\) is sufficiently large. The author stated the following problem in [Sb. Math. 189, No. 7, 955–976 (1997; Zbl 0957.53042)]: For a given compact set \(A\), determine \(r(A)=\inf\{r>0:A\in{\mathcal P}(B_r)\}\), where \({\mathcal P} (B_r)\) is the totality of all Pompeiu sets on \(B_r\). Some upper estimates for \(r(A)\) have been found by C. A. Berenstein, R. Gay and A. Yger [J. Anal. Math. 54, 259–287 (1990; Zbl 0723.44002)]. Here the author determines \(r(A)\) for all ellipsoids \(A\) that are not balls. He then applies his techniques to solving other problems concerning functions with vanishing integrals over ellipsoids.
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
26B15 Integration of real functions of several variables: length, area, volume
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions