Properties Of Holomorphic Functions And The Related Operators In Clifford Analysis

Clifford analysis is one of the main branches in function theory,which studies various analytical properties,functional properties,operator theory and various boundary value problem of functions defined on a subset of real(or complex)high dimension Euclidean space with values in Clifford algebra,which is noncommutative to multiplication.It is a natural promotion of the classical complex analysis with one variable,quaternion analysis and complex analysis with several variables in higher dimensional space.It is one of the hottest problems in Clifford analysis to study the properties of vari-ous holomorphic functions under the Euclidean and non-Euclidean measures and various integral operators related to holomorphic functions.The main object of complex analysis with one variable and with several variables is a holomorphic function.The corresponding function of the holomorphic function in Clifford analysis is called a regular function.It's well known that Cauchy integral formula plays an important role in the classical complex analysis with one variable and with several variables.It is the integral expression of the holomorphic function as well as an important tool to study various properties of the holomorphic function.Its value in the domain can be expressed by the boundary value,and it has important theoretical significance and application value.Cauchy integral formula in Clifford analysis also plays a vital role and is a powerful tool to study the properties of the regular function and to provide us with various integral operators whose good properties play a vital role in the case of solving partial differential equations.At present,few related theoretical research has been done about complex Clifford analysis.This paper presents Cauchy integral formula of the right regular functions in complex Clifford analysis and some functional properties of the related operators which establishes theoretical basis for solving complex partial differential equations.In addition,many academic achievements have been made in the research of regular functions and the related function theory under the Euclidean measures in real Clifford analysis.However,it is still in the active period as to the research for the hypermonogenic and bihypermonogenic function theory under non-Euclidean measures.In this disserta-tion we give a series of properties of some integral operators with the bihypermonogenic kernel.This dissertation is divided into three parts:The first part is the introduction,which gives a survey to the history and academic significance,current state of real and complex Clifford analysis and some main results of this dissertation.The second part consists of the first two chapters,which mainly studies the Cauchy integral formula of right regular functions in complex Clifford analysis,and a series of properties of the T operator with Bochner-Martinelli kernel in complex Clifford analysis.In chapter 1,we first give the Stokes formula in complex Clifford analysis,and then cleverly construct a differential form which contains non-commutative Clifford base ele-ments and complex coordinates,and construct Cauchy integral kernel in complex Clifford analysis.We also obtain Cauchy-Pompieu integral formula and Cauchy integral formula of complex right regular functions by Stokes formula in complex Clifford analysis and Stokes formula in complex variables analysis with several variables.In chapter 2,we define T operator with Bochner-Martinelli kernel in complex Clifford analysis and obtain the boundedness of this operator in Lp space.Then we prove an inequality similar to Hile lemma.We also prove that T operator with Bochner-Martinelli kernel satisfies H(?)lder continuity,?-integrability using this inequality,H(?)lder inequality and Hadamard lemma.The third part consists of chapter 3 and chapter 4,which mainly studies the proper-ties of several singular integral operators in the bounded domain and various boundaries of the integral domain with bihypermonogenic kernel in real Clifford analysis.In chapter 3,we give the definition of T operator with bihypermonogenic kernel in real Clifford analysis and discuss a series of properties of this operator,for instance,the boundedness of this operator in Lp space is obtained,H(?)lder continuity and ?-integrability is also obtained.These properties of T-operator play an important role in solving differ-ential equations.In chapter 4,we define several singular integral operators with bihypermonogenic kernel.For these operators we obtain H(?)lder continuity in the inter of the domain,non characteristic boundary and characteristic boundary.In addition,we obtain Privalov theorem of Cauchy integral operator with bihypermonogenic kernel.