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 : R 2n → Cn of the so-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 C 1 -smooth to Lipschitz. As an illustration, we give a complete viewpoint on simplified proofs of Kytmanov-Aronov-Aȋzenberg type theorems for the case n = 2.We will denote by {e 1 , . . . , e m } an orthonormal basis of the Euclidean space R m . Let C m be the complex Clifford algebra constructed over R m . The noncommutative multiplication in C m is governed by the rules:The Clifford algebra C m is generated additively by elements of the formwhere A = {j 1 , . . . , j k } ⊂ {1, . . . , m} is such that j 1 < · · · < j k , and so the dimension of C m is 2 m . For A = ∅, e ∅ = 1 is the identity element.