Some properties of a T operator with B-M kernel in the complex Clifford analysis

Teodorescu operator, or T-operator, plays an important role in Vekua equation systems and the generalized analytic function theory. It is a generalized solution to the nonhomogeneous Dirac equation. The properties of T operators play a key role in the study of boundary value problems and integral representation of the solutions. In this paper, we first define a Teodorescu operator with B-M kernel in the complex Clifford analysis and prove the boundedness of this operator. Then we give an inequality similar to the classical Hile lemma about real vector which plays a key role in the following proof. Finally, we prove the Hölder continuity and γ-integrability of this operator.


Introduction
In some way, there are two branches of Clifford analysis. The first one is the real Clifford analysis introduced by Brack, Delanghe, and Sommen in [1] which studied function theory with values in a real Clifford algebra defined on a nonempty subset of the Euclidean space R n+1 . Many important theoretic results, such as the Cauchy integral formula, the Cauchy theorem, the Taylor and the Laurent series expansion, the Liouville theorem, and the Morera theorem, have been obtained, and they are the extensions of the well-known classical theorems in one complex variable. Beyond these, a lot of scholars have studied many properties of function theory in the real Clifford analysis. Eriksson and Leutwiler [2][3][4][5] introduced the hypermonogenic function and studied some properties of it. Huang [6], Qiao [7][8][9], Xie [10][11][12], and Yang [13][14][15] obtained many results in Clifford analysis.
The second one is the complex Clifford analysis. In the early 1990s, Ryan [16][17][18][19] introduced the definition of the complex regular function and obtained the Cauchy integral formula whose method is similar to the classical function with one complex variable. In recent years, Ku, Du [20,21] obtained some properties of complex regular functions using the isotonic function.
Based on the above theoretical study and practical background, we construct an analogue of Bochner-Martinelli kernel in several complex variables. We first define a Teodorescu operator with B-M kernel in the complex Clifford analysis and prove the boundedness of this operator. Then we give an inequality similar to the classical Hile lemma about real vector which plays a key role in the following proof. Finally, we prove the Hölder continuity and γ -integrability of this operator.
The basis in Clifford algebra satisfies Define the norm of Clifford numbers as follows: Let ⊂ C n+1 be an open connected nonempty set. Then the function which is defined on and valued in Cl 0,n (C) can be expressed as Dirac operators are introduced as follows [6]: (1) f A (z) is a holomorphic function for any z j ∈ , (2) D l f (z) = 0, ∀z ∈ , then f (z) is called a complex left regular function on . (1) f A (z) is a holomorphic function for any z j ∈ , (2) D r f (z) = 0, ∀z ∈ , then f (z) is called a complex right regular function on . Lemma 2.1 (Hadamard lemma [22]) Let ⊂ R n+1 be a bounded domain, n ≥ 2. If α, β satisfy 0 < α, β < n + 1, and α + β > n + 1, then for any x 1 , x 2 ∈ R n+1 , x 1 = x 2 , we have where J 1 is a positive constant related to α and β.
The notations used in this paper are as follows: (1) ω 2n+2 represents the surface area of unit sphere in a 2n + 2-dimensional real Euclidean space.
. . , 16} are constants only related to n and the size of domain in this paper.

Some properties of a T operator with B-M kernel in the complex Clifford analysis
In this section, we discuss some properties of a singular integral operator.
Remark 1 Because the original Hile lemma cannot be used directly in the complex Clifford analysis, we give the conclusion of Theorem 3.2 which is similar to the classical Hile lemma and plays an important role in proving the properties of T-operators and Cauchy operators. We insert the appropriate items according to the situation and prove that inequality (2) holds. Inequality (2) is similar to the Hile lemma of the classical real vector and is complete symmetry with respect to the variable ξ , z. It is a good tool to prove the Hölder continuity of the T operator with B-M kernel in the complex Clifford analysis. Theorem 3.3 Let ⊂ C n+1 be a bounded domain, ϕ ∈ L p ( ), p > 2(n + 1), then for any z 1 , z 2 ∈ , we have and Tϕ is Hölder continuous on , where α = 1 -2(n+1) p .
From Hadamard's lemma, we get So Hence Using Hölder's inequality, we obtain Remark 2 In Case 2 of this theorem, we use the inequality of Theorem 3.3, Hölder's inequality, and Hadamard's lemma. This result enriches the theoretical system of the complex Clifford analysis.