Extension and tangential CRF conditions in quaternionic analysis

TL;DR

This paper establishes extension theorems for quaternionic holomorphic functions, demonstrating how functions satisfying certain tangential conditions can be locally extended, and discusses implications for the Dirichlet problem and octonionic cases.

Contribution

It introduces new extension theorems for quaternionic holomorphic functions with tangential conditions, including solutions for the Dirichlet problem and extensions to continuous functions.

Findings

Existence of local extensions for quaternionic holomorphic functions

Solution to the Dirichlet problem with smooth data

Extension of results to continuous functions and discussion of octonionic case

Abstract

We prove some extension theorems for quaternionic holomorphic functions in the sense of Fueter. Starting from the existence theorem for the nonhomogeneous Cauchy-Riemann-Fueter Problem, we prove that an $\mathbb{H}$-valued function $f$ on a smooth hypersurface, satisfying suitable tangential conditions, is locally a jump of two $\mathbb{H}$-holomorphic functions. From this, we obtain, in particular, the existence of the solution for the Dirichlet Problem with smooth data. We extend these results to the continous case. In the final part, we discuss the octonian case.