Function Spaces in Quaternionic and Clifford Analysis

The purpose of this chapter is to describe the settings in which one can set up coherent generalizations to several variables of the classical single variable complex analysis and to point out some basic properties of the function spaces that naturally occur in such settings. Particular emphasis is placed on defining proper counterparts in higher dimension of the concept of holomorphic functions and of the Cauchy–Riemann operator. The resulting function theory deals with monogenic functions and Euclidean Dirac or Cauchy–Riemann operators. Its richness is illustrated by the existence of a general Cauchy–Pompeiu Integral Representation Formula. Actually, even more general formulas will be initially derived for first order homogeneous differential operators with constant coefficients in a unital associative Banach algebra. The point that will be made is that the operators and algebras that yield the simplest forms of such Integral Representation Formulas with Remainders are essentially the Cauchy–Riemann and Dirac operators associated with Euclidean Clifford algebras. As yet another generalization of a classical result in single variable complex analysis one gets a Quantitative Hartogs–Rosenthal Theorem concerned with uniform approximation on compact sets by monogenic functions. An equally important goal of the chapter is to outline some of the contributions that led to the discovery and development of Quaternion and Clifford analysis. The last section includes several concluding remarks and a few additional references that outline the full scope of some of the early and current developments, and help in further explorations of other lines of research in quaternion and Clifford analysis.