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

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.