Poincaré-Bertrand formula for Martinelli-Bochner integral
The formula in question tells how to interchange the order of integration in an iterated singular integral: \[ \int_{\partial D_ w} U(w,z) \int_{\partial D_ \zeta} f(\zeta,w) U (\zeta, w) = {1 \over 4} f(z,z) + \int_{\partial D_ \zeta} \int_{\partial D_ w} f(\zeta,w) U(w,z) U(\zeta, w), \] where \(z\) is a point in the boundary of a bounded domain \(D\) in \(\mathbb{C}^ n\) with class \(C^ 1\) boundary, \(U\) is the Bochner- Martinelli kernel, \(f\) is a Hölder continuous function on \(\partial D \times \partial D\), and the integrals are understood in the sense of principal values. This Poincaré-Bertrand formula is due to \textit{A. I. Serbin} [Izv. Vyssh. Uchebn. Zaved., Mat. 1973, No. 12(139), 64-72 (1973; Zbl 0281.32001)]. The authors point out that Serbin's proof is invalid, for it is based on an incorrect inversion formula for the singular Bochner-Martinelli integral given in a previous paper of \textit{A. I. Serbin} [Sov. Math., Dokl. 12, 357-361 (1971); translation from Dokl. Akad. Nauk SSSR 196, 1276-1279 (1971; Zbl 0229.32003)]. The authors give an explicit counterexample in the unit ball to show that Serbin's inversion formula is false when \(n > 1\). Nonetheless, the Poincaré-Bertrand formula is true, and the authors supply a valid proof.
scientific article; zbMATH DE number 640193