Octonion analysis of several variables (Q2255035)

Let \(\mathbb{O}\) be the non-commutative, non-associative, normed division real algebra of Cayley's octonions. The non-associativity of multiplication in \(\mathbb{O}\) causes obvious difficulties in attempts to construct an analysis of \(\mathbb{O}\)-valued functions but nevertheless the analogues of some basic facts of one-dimensional complex analysis are known for octonion-valued functions of one octonionic variable. Here, the authors show that some basic facts of multidimensional complex analysis have their exact images for \(\mathbb{O}\)-valued functions of several octonionic variables. More precisely, they introduce the Bochner-Martinelli kernel in several octonionic variables and establish the corresponding integral formula; they consider the octonionic analogue of the \(\overline{\partial}\)-problem with compact support (the non-homogeneous octonionic Cauchy-Riemann equations) and they deduce Hartogs' theorem for several octonionic variables. The proofs follow closely the lines of their antecedents from complex, quaternionic and Clifford analysis without the necessity of attracting new ideas.