Slice Analysis In High Dimension

Slice analysis is the generalization of function theory of one complex variable to the non-commutative,non-associative fields,and it has been fully developed after more than a decade of research.However,since the non-commutativity of the complex struc-tures,it is difficult to generalize the theory of several complex variables to the slice theory.In order to effectively solve the problem,we propose to go in two steps.That is,we first establish the analysis theory on the slice with the same complex structures,then we establish the representation formula and use it to lift the theory to the whole space to set up the several variables slice analysis theory.It consists of four chapters.The first chapter is an introduction to the background of our research field,and research status,as well as the basic results and research methods of slice analysis in several variables.The second chapter we establish the slice analysis in several octonionic variables,which is the theory of slice regular functions theory in several octonionic variables with the same complex structures.We obtain the corresponding Splitting lemma of the sev-eral octonionic variables slice theory.This lemma is particularly important in slice anal-ysis,which enables us to generalize some classical results in several complex variables into slice analysis.We establish the Bochner-Martinelli formula of several octonionic variables slice functions,as well as the corresponding Hartogs extension theorem.In the third chapter,we study the geometric function theory of the slice regular functions of several variables.It involves the slice mapping and the corresponding slice regular mapping of several variables in quadratic cones of Clifford algebras.We prove that if a slice map f preserves a slice,then its maximum modulus as well as minimum modulus can achieve.The point is that this property holds without assumption that f is slice regular.On the unit ball on the quadratic cones of Clifford algebras,we establish the growth theorem for slice regular starlike or convex maps.This result can even be generalized to the bounded slice domains which are slice starlike and slice circular.In the fourth chapter,we study function theory of the slice regular functions of several variables.On the weighted Bergman space of slice regular functions of several quaternionic variables,we study the properties of the related operators,the integral rep-resentation formula on a slice of the unit ball,Bergman projection and an equivalent characterization of the weighted Bergman space.We establish an integral representa-tion formula on the whole unit ball of the weighted Bergman space for slice regular functions of several quaternionic variables.