On functions that are representable by the Cauchy-Fantappiè integral of a certain form (Q2716201)

The authors prove the following theorem: Let \(D\) be a bounded domain in \(\mathbf C^n\) with connected boundary \(\partial D\) of class \(C^{[\gamma]+1}\), and \(f\in C^\gamma(\partial D)\), \(\gamma>0\), a function satisfying \(\int_{\partial D} f(\zeta) U_\alpha(\zeta,z)=0\) \(\forall z\notin\overline D\), where NEWLINE\[NEWLINE U_\alpha(\zeta,z) = c_n \frac{\prod_{j=1}^n (\alpha_j|\zeta_j-z_j|^{2\alpha_j-2}) \sum_{k=1}^n (-1)^{k-1} \frac{\overline\zeta_k-\overline z_k}{\alpha_k} d\overline\zeta[k] \wedge d\zeta} {(\sum_{j=1}^n |\zeta_j-z_j|^{2\alpha_j})^n} NEWLINE\]NEWLINE where \(d\zeta=d\zeta_1\wedge\dots\wedge d\zeta_n\), \(d\overline\zeta[k] =d\overline\zeta_1\wedge\dots\wedge d\overline\zeta_{k-1}\wedge d\overline\zeta_{k+1}\wedge\dots\wedge d\overline\zeta_n\), \(c_n=\frac{(n-1)!}{(2\pi i)^n}\), and \(\alpha_1,\dots,\alpha_n\) are natural numbers. Then \(f\) extends holomorphically into \(D\) to a function \(F\in C^\gamma(\overline D)\). (Here \([\gamma]\) is the integer part of \(\gamma\) and \(C^\gamma\) stands for the space of functions in \(C^{[\gamma]}\) whose derivatives of order \([\gamma]\) are Hölder continuous of order \(\gamma-[\gamma]\).) For \(\alpha_1=\dots =\alpha_n=1\), when the form \(U_\alpha(\zeta,z)\) reduces to the Bochner-Martinelli kernel \(U(\zeta,z)\), this result was obtained in several earlier papers of which the first present author was a coauthor; and for \(n=2\) the theorem was proved by the present authors in [``On a criterion for existence of holomorphic extension of functions in \(\mathbf C^2\)'', Multidimensional complex analysis, Krasnoyarskij Gosudarstvennyj Universitet, Krasnoyarsk, 78-92 (1994)].NEWLINENEWLINEFor the entire collection see [Zbl 0959.00006].