Holomorphic Extension Theorems in Lipschitz Domains of $${\mathbb{C}}^{2}$$

Abstract

The holomorphic functions of several complex variables are closely related to the continuously differentiable solutions \(f : {\mathbb{R}}^{2n} \mapsto {\mathbb{C}}_{n}\) of the so called isotonic system $$\partial _{\underbar{x}_1 } + i \tilde{f} \mathop{\partial _{\underbar{x} _2 } = 0}$$ . The aim of this paper is to bring together these two areas which are intended as a good generalization of the classical one-dimensional complex analysis. In particular, it is of interest to study how far some classical holomorphic extension theorems can be stretched when the regularity of the boundary is reduced from C1-smooth to Lipschitz. As an illustration, we give a complete viewpoint on simplified proofs of Kytmanov-Aronov-Aĭzenberg type theorems for the case n = 2.