The Properties Of The Isotonic Type T-operator In Clifford Analysis

Isotonic Clifford analysis is a new and active branch in Clifford analysis, which studies the holomorphic function denned on the even dimensional Euclidean space R2m and valued in the complex Clifford algebra Co,m.It is also regarded as an elegant generalization of the complex Clifford algebra of holomorphic functions of several complex variables, and it has important theoretical significance. In this paper, the properties of the Isotonic type T-operator are studied, which is a kind of singular integral operator related to the holomorphic function in Isotonic Clifford analysis.In Isotonic Clifford analysis, the Cauchy type integral of the Isotonic functions and its boundary characteristics are well developed. However, Isotonic type T-operator and its properties haven’t been studied. In this paper, the properties of this operator in Isotonic Clifford analysis are discussed.In chapter 1, the basic structure and operation laws of the complex Clifford algebra are introduced and some important lemmas are given.In chapter 2, first, the definition of the Isotonic type T-operator TΩiso which is defined on a bounded domain is given, then the boundedness of the operator on space LP(Ω, Co,m) is discussed, finally the Holder continuity on R2m and the γ times integrability on bounded domain Ω∈ R2m of (TΩios[g])(x) are studied.In chapter 3, first, two important inequalities are proved, which play a key role in the following proofs, then on the basis of the second chapter, the Isotonic type T-operator Tiso on the unbounded domain is deined through the method of inverse transformation, finally two fundamental properties of (Ttso[g]) (x) on R2m are discussed, which are the boundedness and Holder continuity.