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.