The Hartogs-Bochner extension for monogenic functions of several vector variables and the Dirac complex

TL;DR

This paper extends the Hartogs-Bochner theorem to monogenic functions of multiple vector variables, utilizing the Dirac complex and PDE methods to analyze boundary behavior and extension properties of these functions.

Contribution

It provides a simple characterization of the first four spaces in the Dirac complex and establishes the Hartogs-Bochner extension for tangentially monogenic functions.

Findings

Explicit description of the first three operators in the Dirac complex.

Proof of ellipticity of the initial part of the Dirac complex.

Extension of monogenic functions across boundaries via Hartogs-Bochner theorem.

Abstract

Holomorphic functions in several complex variables are generalized to regular functions in several quaternionic variables, and further to monogenic functions of several vector variables, which are annihilated by several Dirac operators on $k$ copies of the Euclidean space $\mathbb R^n$. As the Dolbeault complex in complex analysis, the Dirac complex resolving several Dirac operators plays the fundamental role to investigate monogenic functions. Although the spaces in the Dirac complex are complicated irreducible modules of ${\rm GL}(k),$ we give a simple characterization of the first four spaces, which allows us to write down first three operators in the Dirac complex explicitly and to show this part to be an elliptic complex. Then the PDE method can be applied to obtain solutions to the non-homogeneous several Dirac equations under the compatibility condition, which implies the Hartogs' phenomenon for monogenic functions. Moreover, we find the boundary version of several Dirac operators and introduce the notion of a tangentially monogenic function, corresponding to tangential Cauchy-Riemann operator and CR functions in several complex variables, and establish the Hartogs-Bochner extension for tangentially monogenic functions on the boundary of a domain.