The Bochner–Martinelli integral operator for real analytic functions

Let $D$ be a bounded domain in $\mathbb C^n$ ($n>1$) with a real analytic connected boundary $\partial D=\Gamma$. The Bochner–Martinelli integral (integral operator) $M(f)$ is considered for real analytic functions $f$ on $\Gamma$. It is shown that the integral $M(f)$ is real analytic up to $\Gamma$. Iterations of the Bochner–Martinelli integral $M^k(f)$ are considered. It is proved that they converge to a function holomorphic in $\overline{D}$ at $k\to\infty$. The Bochner–Martinelli transform $M(T)(z)$ is defined for analytical functionals $T$. It is proved that the iterations of $M^k(T)(z)$ converge weakly to a $CR$-functional at $k\to\infty$.