| Clifford algebra is a noncommutable and associative algebra created by W.K Clifford, which is created on the geometric algebra theory. Clifford analysis mainly discuss the functions defined in the Euclidean spaces Rn+1, and valued from Clifford algebra An(R). It can be regarded as the high-dimensional generalization of the real analysis, complex analysis, quaternionic analysis etc. Clifford analysis has developed into an active branch of the mathematical study, which has brought significant theories and applicable values in mathematics and other subjects. The Cauchy type integral operator and the Cauchy type integral operator for density function containing parametor in Clifford analysis and complex analysis are studied. Based on this, this paper studies the quasi-Cauchy type integral operator for density function containing parametor in its hyper-monogenic kernel. And we also get the Holder continuity and the Plemelj formnula of the quasi-Cauchy type integral operator for the density function containing parametor in its hyper-monogenic kernel. Futhermore, the boundedness and the γ-time integrability of quasi-Cauchy type integral operator with hyper-monogenic kernel in weighted Lebesgue spaces are proved in Clifford analysis.This paper is made up of three parts. In the first part, the preliminaries and some main lemmas and definitions of the quasi-Cauchy type integral operator are given. Then, in the second part, we give the definition of the quasi-Cauchy type integral operator for density function containing parameter in its hyper-monogenic kernel, we discuss the Holder continuity of this operator and its Plemelj formnula. In the third part, we give the boundedness and the γ-time integrability of quasi-Cauchy type integral operator in weighted Lebesgue spaces.The proceeding contents have developed the theory of hypermonogenic functions, all of these contents lay a solid foundation for studying the properties and applications of hypermonogenic functions. |
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