Gilbert's conjecture and A new way to octonionic analytic functions from the clifford analysis

TL;DR

This paper affirms Gilbert's conjecture on Hardy spaces of Clifford analytic functions in 8D, using octonion algebra to construct spinor and Clifford spaces, leading to a new associative framework for octonionic analysis.

Contribution

It provides an explicit construction of Spinor and Clifford spaces via octonions, establishing an associative approach to octonionic analytic functions and confirming Gilbert's conjecture.

Findings

Confirmed Gilbert's conjecture on Hardy spaces in 8D

Constructed Spinor and Clifford spaces using octonions

Reformulated classical results on octonionic analytic functions

Abstract

In this article we will give a affirmative answer to Gilbert's conjecture on Hardy spaces of Clifford analytic functions in upper half-space of $\mathbb{R}^8$. It depends on a explicit construction of Spinor space $\mathcal{R}_8$ and Clifford algebra $Cl_8$ by octonion algbra. What's more , it gives us an associative way to octonionic analytic function theory. And the similar question has been discussed in Octonionic Hardy space in upper-half space, some classical results about octonionic analytic functions have been reformulated, too.