Quaternionic and Clifford Analysis in Several Variables

This article discusses how the theory of Fueter regular functions on quaternions can be extended to the case of several variables. This can be done in two different (complementary) ways. One can follow the traditional approach to several complex variables developed in the first part of the twentieth century, and construct suitable generalizations of the Cauchy–Fueter formula to the setting of several variables. In this way one obtains an analog of the Bochner–Martinelli formula for regular functions of several quaternionic variables, and from that starting point one can develop most of the fundamental results of the theory. On the other hand, one can take a more algebraic point of view, in line with the general ideas of Ehrenpreis on solutions to systems of linear constant coefficients partial differential equations, and exploit the fact that regular functions in several variables are infinitely differentiable functions that satisfy a reasonably simple overdetermined system of differential equations. By using this characterization, and the fundamental ideas pioneered by Ehrenpreis and Palamodov, one can construct a sheaf theoretical approach to regular functions of several quaternionic variables that rather immediately allows one to discover important global properties of such functions, and indeed to develop a rigorous theory of hyperfunctions in the quaternionic domain. This article further shows how this process can be adapted to variations of Fueter regularity such as biregularity and Moisil–Theodorescu regularity, as well as to the case of monogenic functions of several vector variables. Quaternion(s) Finally the article considers the notion of slice monogeneity and slice regularity, and shows how they can also be extended to several variables. The theories in these cases are very recent, and rapidly developing.