Asymptotically holomorphic functions and their applications. (Russian) Zbl 0639.32005

Pinchuk, S. I.; Khasanov, S. V.

Mat. Sb., N. Ser. 134(176), No. 4(12), 546-555 (1987).

The authors weaken the notion of holomorphic function in \({\mathbb{C}}^ n\) by requiring only a certain bound on \({\bar \partial}\) near a totally real submanifold. The main results proved here concerning such “asymptotically holomorphic” functions are (1) a version of the edge of the wedge theorem, (2) a generalization of a classical theorem of E. Lindelöf to describe the behaviour of asymptotically holomorphic functions under non-tangential approach to the boundary, (3) the extension of certain holomorphic mappings from certain wedge-like domains to the edge, (4) the extension to the boundary of arbitrary proper holomorphic mapping of strictly pseudoconvex domains in \({\mathbb{C}}^ n\); this latter result is an improvement of earlier results due to the first author, Sib. Mat. Zhur. 15, 909-917 (1974; Zbl 0289.32011), G. M. Khenkin, Sov. Math., Dokl. 14, 858-862 (1973); translation from Dokl. Akad. Nauk SSSR 210, 1026-1029 (1973; Zbl 0288.32015), Ch. Fefferman, Invent. Math. 26, 1-65 (1974; Zbl 0289.32012), L.Lempert, Bull. Soc. Math. France 109, 427-474 (1981; Zbl 0492.32025), L. Nirenberg, S. Webster and P. Yang, Commun. Pure Appl. Math. 33, 305-338 (1980; Zbl 0436.32018), and others.

Reviewer: E.J.Akutowicz

MSC:

32A40	Boundary behavior of holomorphic functions of several complex variables
32V40	Real submanifolds in complex manifolds

Keywords:

boundary behaviour; edge of the wedge; asymptotically holomorphic functions; strictly pseudoconvex