On some conditions for the existence of a holomorphic continuation of functions in a ball

The paper considers the Cauchy–Fantappiè integral representation, which is close to the Bochner–Martinelli integral representation, and the kernel of which consists of derivatives of the fundamental solution of the Laplace equation. The aim of the work is to study the properties of this integral representation for integrable functions. Namely, the paper considers an integral (integral operator) with this kernel for integrable functions $f$ on the boundary $S$ of the unit ball $B$. Iterations of the integral of this integral operator of the order $k$ are considered. We prove that they converge to a function holomorphic in $B$ as $k\to\infty$.