On iterations of the Cauchy-Fantappié transform of analytic functionals

We consider the Cauchy-Fantappié integral representation of a certain form $Q[f]$ for real analytic functions $f$ on the boundary of a bounded domain $D$ with real analytic connected boundary $\Gamma$ in the complex space $\mathbb C^n, \ n>1$ . Its kernel consists on derivatives of the fundamental solution of the Laplace equation. Previously, the authors considered iterations of this integral operator $Q^m[f]$ for smooth functions $f$ and showed that they converge to a holomorphic function as $m\to \infty$.
Here we define the Cauchy-Fantappié transform $Q[T](z)$ for analytic functionals $T$. We prove that the iterations $Q^m[T](z)$ converge to the $CR$-functional as $m\to\infty$.
This is a joint work with A. M. Kytmanov, Siberian Federal University, Krasnoyarsk.
Authors was supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement No. 075-02-2025-1606).