Complex Interpolation And K-hypermonogenic Functions In Clifford Algebras

The generalized Cauchy-Riemann equation,Cauchy integral formula and in-terpolation theory are classical problems in harmonic analysis,which have impor-tant applications in analysis and partial differential equations.Clifford algebra,also known as Geometric algebra,is a generalization of quaternion algebra,outer algebra and complex algebra,integrating inner and outer product operations.It is widely used in geometry and physics.Many scholars have conducted a great deal of research on Clifford algebra,among which the famous ones are Hamilton(quaternion),Grassmann(outer algebra),Clifford,Hestenes and so on.Clifford algebra is also widely used in other fields such as general relativity,quantum mechanics,quantum field theory,projective geometry,differential geometry and conformal geometry.This dissertation focuses on three aspects of Clifford algebra:complex interpolation in quaternion valued function spaces,solutions for general-ized Cauchy-Riemann systems of two-sided dualk-hypermonogenic functions and its Cauchy-type integral formulas.We obtain Riesz-Thorin interpolation theorem inL^P(C,H),solutions for Vekua-type systems of two-sided dualk-hypermonogenic functions and the Cauchy-type integral formulas for dual(1-m)-hypermonogenic functions.They enriched the theory of Clifford algebra.In chapter 1,we introduce the research background of complex interpolation theorem in quaternion valued function spaces,the research status and significance of the three problems.In Chapter 2,we review the basic knowledge of Clifford analysis,includingk-vector subspaces of real Clifford algebra and operation properties of elements in Clifford algebra.In chapter 3,we recall the theory of monogenic functions.Firstly,the defi-nitions of left,right and two-sided monogenic functions are introduced.Then we assert that the axial monogenic function can be expressed in homogeneous mono-genic polynomial of orderk?N₀,where the coefficient functions satisfy Vekua-type systems.Finally,the transforms of axial two-sided monogenic functions and axial left monogenic functions are presented,that is,theorem 3.1.3.In chapter 4,we prove the complex interpolation theorem in the quaternion valued function spaces.On one hand,the Riesz-Thorin interpolation theorem inL^P(C,H)and its two applications are proposed.On the other hand,we give the extension of the first and second complex interpolation theory inL^P(C,H)by using Calder?on complex interpolation methods.In chapter 5,we present solutions for the generalized Cauchy-Riemann sys-tems of two-sided dualk-hypermonogenic functions,and characterize the two-sided dualk-hypermonogenic functions by a refinement for its definition.In chapter 6,we propose the Cauchy-type integral theorem of dualkhyper-monogenic functions.Then the Cauchy-type integral formulas of dual(1-m)-hypermonogenic functions are presented.Finally,we discuss the Hilbert transfor-mations of hypermonogenic functions on the unit sphere in R^m+1.