Separate holomorphic extension of CR functions (Q1014799)

The results in the paper are a generalization of the classical Hartogs' theorem on separate holomorphicity [\textit{F. Hartogs}, Math. Ann. 62, 1--88 (1906; JFM 37.0444.01)]: A holomorphic function on a domain of \(\mathbb C^n\) which is holomorphic along each coordinate line is holomorphic. A first generalization is given by replacing the coordinate lines in the Hartogs' hypothesis by a \(2(n-m)\)-parameters family of holomorphic \(m\)-leaves \(\mathcal F\) having \(\mathcal C^\omega\) dependence on parameters. A second generalization is given by starting from a CR function \(f\) on a generic \(m\)-codimensional manifold \(M\) transversal to the foliation \(\mathcal F\) having holomorphic extension on each leaf (notice that \(\mathcal C^0\) gluing of extensions is not required). Then the function \(f\) actually extends holomorphically to the domain swept by the leaves of \(\mathcal F\). Other results of this flavour are obtained reducing the parameter space (i.e., not requiring it to be an open subset of \(\mathbb C^{n-m}\)) generalizing a result of \textit{L. A. Ajzenberg} and \textit{C. Rea} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 20, No.~2, 313--322 (1993; Zbl 0787.32021)]. The basic tool developed and used in the paper is an estimate on subharmonic functions of one complex variable relying on Fatou's lemma and the Phragmén-Lindelöf principle.