On holomorphic continuation of functions along boundary sections

Imomkulov, S. A. ; Khujamov, J. U.

Mathematica Bohemica, Tome 130 (2005) no. 3, pp. 309-322.

Résumé

Let $D^{\prime } \subset \mathbb{C}^{n-1}$ be a bounded domain of Lyapunov and $f(z^{\prime },z_n)$ a holomorphic function in the cylinder $D=D^{\prime }\times U_n$ and continuous on $\bar{D}$. If for each fixed $a^{\prime }$ in some set $E\subset \partial D^{\prime }$, with positive Lebesgue measure $\text{mes}\,E>0$, the function $f(a^{\prime },z_n)$ of $z_n$ can be continued to a function holomorphic on the whole plane with the exception of some finite number (polar set) of singularities, then $f(z^{\prime },z_n)$ can be holomorphically continued to $(D^{\prime }\times \mathbb{C})\setminus S$, where $S$ is some analytic (closed pluripolar) subset of $D^{\prime }\times \mathbb{C}$.

Détail

DOI : 10.21136/MB.2005.134101

Classification : 32D15, 46G20
Keywords: holomorphic function; holomorphic continuation; pluripolar set; pseudoconcave set; Jacobi-Hartogs series