Gilbert’s Conjecture and a New Way to Octonionic Analytic Functions from the Clifford Analysis

In this article, we will give an affirmative answer to Gilbert’s conjecture on Hardy spaces of Clifford analytic functions in upper half-space of \(\mathbb {R}^{8}\) ${\mathbb{R}}^8$ . It is based on an explicit construction of Clifford algebra \(Cl_8\) $C{l}_8$ and Spinor space \(\mathcal {R}_{8}\) ${\mathcal{R}}_8$ by octonion algebra. Furthermore, it provides an associative approach to the theory of octonionic analytic functions. Additionally, certain classical results about octonionic analytic functions have been reformulated, and a related subject has been treated in Octonionic Hardy space in upper half-space.