The area integral and boundary values of octonion-valued monogenic functions

TL;DR

This paper introduces the area integral for octonion-valued monogenic functions, establishing boundary value existence and limits, and addressing non-commutativity and non-associativity issues using Stein's method and subharmonic functions.

Contribution

It develops a new area integral framework for octonion monogenic functions and proves boundary value existence, handling octonion algebra complexities with Stein's method.

Findings

Non-tangential boundary values exist almost everywhere.

Normal boundary values exist at specified boundary points.

Limits of scalar and other components are either all existing or all not.

Abstract

We introduce the Area Integral for octonion-valued monogenic functions in the half-space. It is used to prove the existence of the non-tangential boundary values almost everywhere and of the normal boundary values at a given boundary point for these classes of functions. It is also proved that the non-tangential limit for the scalar component of a monogenic function and those limits for all of its other seven components exist or not exist simultaneously. It turns out that non-commutativity and non-associativity of octonions in certain problems can be circumvented by making use of P. Stein method and employing subharmonic functions.