Skip to content

Advertisement

  • Research
  • Open Access

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

Journal of Inequalities and Applications20182018:226

https://doi.org/10.1186/s13660-018-1816-6

  • Received: 6 July 2018
  • Accepted: 15 August 2018
  • Published:

Abstract

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.

Keywords

  • Complex Clifford analysis
  • Teodorescu operator
  • Boundedness
  • Hölder continuity
  • γ-integrability

1 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 [25] introduced the hypermonogenic function and studied some properties of it. Huang [6], Qiao [79], Xie [1012], and Yang [1315] obtained many results in Clifford analysis.

The second one is the complex Clifford analysis. In the early 1990s, Ryan [1619] 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.

2 Preliminaries

Let \(\mathrm{Cl}_{0,n}(C)\) be a complex Clifford algebra over n+1-dimensional Euclidean space \({\mathbf{C}}^{n+1}\). \(\mathrm{Cl}_{0,n}(C)\) has the basis \(e_{0}, e_{1}, e_{2}, \ldots, e_{n}; e_{1}e_{2}, e_{1}e_{3}, \ldots, e_{1}e_{n}; e_{2}e_{3}, \ldots, e_{2}e_{n}; \ldots; e_{n-1}e_{n}; \ldots; [4] e_{1}\cdots{e_{n}}\). Hence, an arbitrary element of the basis may be written as \(e_{A}=e_{\alpha_{1}}\cdots e_{\alpha_{h}}\), where \(A=\{\alpha_{1}, \ldots, \alpha_{h} \}\subseteq\{1, \ldots, n\}\), \(1\le\alpha_{1}<\alpha_{2}<\cdots<\alpha_{h}\le n\) and when \(A=\emptyset\), \(e_{A}=e_{0}=1\). So, the complex Clifford algebra is composed of elements having the type \(a=\sum_{A}z_{A}e_{A}\), where \(z_{A}\) are complex numbers.

The basis in Clifford algebra satisfies
$${e_{i}^{2}=-1,\quad i=1, 2, \ldots, n}, \qquad e_{i}e_{j}=-e_{j}e_{i},\quad 1\leq i< j\leq n, (i\neq j). $$
Define the norm of Clifford numbers as follows:
$$\biggl\vert \sum_{A}z_{A}e_{A} \biggr\vert =\sqrt{(a, a)}= \biggl(\sum_{A} \vert z_{A} \vert ^{2} \biggr)^{\frac{1}{2}}. $$
Let \(\Omega\subset\mathbf{C}^{n+1}\) be an open connected nonempty set. Then the function which is defined on Ω and valued in \(\mathrm{Cl}_{0,n}(C)\) can be expressed as \(f(z)=\sum_{A}{f_{A}(z)e_{A}}\), where \(f_{A}(z)\) are complex-valued functions. Let
$${F^{(r)}_{\Omega}=\biggl\{ f\Big|f:\Omega\rightarrow \mathrm{Cl}_{0,n}(C), f(z)=\sum_{A}{f_{A}(z)e_{A}}, f_{A}(z)\in C^{r}(\Omega)\biggr\} }. $$
Dirac operators are introduced as follows [6]:
$$\begin{gathered} D^{l} f=\sum_{i=0}^{n}e_{i} \frac{\partial f}{\partial z_{i}};\qquad \overline{D^{l}}f=e_{0}{ \frac{\partial f }{\partial z_{0}}}-\sum_{i=1}^{n}e_{i}{ \frac{\partial f}{\partial z_{i}}}; \\ D^{r} f=\sum_{i=0}^{n} \frac{\partial f}{\partial z_{i}}e_{i};\qquad \overline{D^{r}}f={ \frac{\partial f }{\partial z_{0}}}e_{0}-\sum_{i=1}^{n}{ \frac{\partial f}{\partial z_{i}}}e_{i}. \end{gathered} $$

Definition 2.1

([16])

If \(\Omega\subset C^{n+1}\), \(f: \Omega\rightarrow \mathrm{Cl}_{0,n}(C)\) satisfies:
  1. (1)

    \(f_{A}(z)\) is a holomorphic function for any \(z_{j}\in\Omega\),

     
  2. (2)

    \(D_{l}f(z)=0\), \(\forall z\in\Omega\),

     
then \(f(z)\) is called a complex left regular function on Ω.

Definition 2.2

([16])

If \(\Omega\subset C^{n+1}\), \(f: \Omega\rightarrow \mathrm{Cl}_{0,n}(C)\) satisfies:
  1. (1)

    \(f_{A}(z)\) is a holomorphic function for any \(z_{j}\in\Omega\),

     
  2. (2)

    \(D_{r}f(z)=0\), \(\forall z\in\Omega\),

     
then \(f(z)\) is called a complex right regular function on Ω.

Lemma 2.1

(Hadamard lemma [22])

Let \(\Omega\subset R^{n+1}\) be a bounded domain, \(n\geq2\). If α, β satisfy \(0<\alpha,\beta <n+1\), and \(\alpha+\beta>n+1\), then for any \(x_{1},x_{2}\in R^{n+1}\), \(x_{1}\neq x_{2}\), we have
$$\int_{\Omega} \vert t-x_{1} \vert ^{-\alpha} \vert t-x_{2} \vert ^{-\beta}\,dt\leq J_{1} \vert x_{1}-x_{2} \vert ^{(n+1)-\alpha-\beta}, $$
where \(J_{1}\) is a positive constant related to α and β.

Lemma 2.2

([22])

Let \(\Omega\subset R^{n+1}\) be a bounded domain, when \(\alpha< n+1\), for any \(y\in R^{n+1}\), we have
$$\int_{\Omega} \vert x-y \vert ^{-\alpha}\,dx\leq M, $$
where M is a positive constant only related to α and the size of Ω.

Lemma 2.3

(Hölder inequality [23])

If \(f_{k}\in L^{p_{k}}(\Omega)\), \(k=1,2,\ldots,n\), and
$${\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}} \leq1,} $$
then \({f_{1}f_{2}\cdots f_{n}\in L^{p}(\Omega)}\), and
$$L^{p}(f_{1}f_{2}\cdots f_{n})\leq L^{p_{1}}(f_{1})L^{p_{2}}(f_{2}) \cdots L^{p_{n}}(f_{n}), \quad p\geq1. $$

Lemma 2.4

(Minkowski inequality [23])

If \({f_{1},f_{2},\ldots,f_{n}\in L^{p}(\Omega)}\), then \(f_{1}+f_{2}+\cdots+f_{n}\in L^{p}(\Omega)\), and
$${L^{p}(f_{1}+f_{2}+\cdots+f_{n})\leq L^{p}(f_{1})+L^{p}(f_{2})+ \cdots+L^{p}(f_{n}), p\geq1.} $$

Lemma 2.5

([23])

Let \(L^{p}(\Omega,\mathrm{Cl}_{0,n}(R))\) represent the set of all p order integrable functions which are defined on the bounded domain \(\Omega \subset R^{n+1}\), and with values in the real Clifford algebra \(\mathrm{Cl}_{0,n}(R)\), define the norm of φ as follows:
$$\Vert \varphi \Vert _{\Omega,p}= \biggl( \int_{\Omega} \bigl\vert \varphi (x) \bigr\vert ^{p}\,dV_{x} \biggr)^{\frac{1}{p}},\quad p\geq1, $$
when \(1\leq r \leq p\),
$$L^{p}\bigl(\Omega, \mathrm{Cl}_{0,n}(R)\bigr)\subset L^{r} \bigl(\Omega, \mathrm{Cl}_{0,n}(R)\bigr) $$
is true.
The notations used in this paper are as follows:
  1. (1)

    \(\omega_{2n+2}\) represents the surface area of unit sphere in a \(2n+2\)-dimensional real Euclidean space.

     
  2. (2)

    \(M_{i}\) \(\{i=1,2,3\}\), \(K_{i}\) \(\{i=1,\dots,16\}\) are constants only related to n and the size of domain Ω in this paper.

     
  3. (3)

    \(dV_{\xi}=d\zeta_{0}\wedge d\zeta_{1}\wedge\cdots\wedge d\zeta_{n}\wedge d\eta_{0}\wedge d\eta_{1}\wedge\cdots\wedge d\eta _{n}\), \(\zeta_{j}\in R\), \(\eta_{j}\in R\), (\(j=0, 1, \ldots, n\)), \(\xi_{j}=\zeta _{j}+i\eta_{j}\).

     
  4. (4)

    \(d\bar{\xi}\wedge d{\xi}=d\bar{\xi}_{0}\wedge d\bar{\xi}_{1}\wedge \cdots d\bar{\xi}_{n}\wedge d{\xi_{0}}\wedge d{\xi_{1}}\dots\wedge d{\xi_{n}}\).

     
  5. (5)

    \(d\bar{\xi}\wedge d{\xi}=(2i)^{n+1}\,dV_{\xi}\).

     

3 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.

Definition 3.1

Let \(\Omega\subset C^{n+1}\) be a bounded domain, \(\varphi\in L^{p}(\overline{\Omega},\mathrm{Cl}_{0,n}(C))\), \(z\in C^{n+1}\), then
$$\begin{aligned} &(T\varphi) (z) \\ &\quad=\frac{1}{\omega_{2n+2}(2i)^{n+1}} \int_{\Omega}\varphi(\xi) \biggl(\frac{\sum_{k=0}^{n}(\xi_{k}-z_{k})\bar{e}_{k}}{ \vert \xi-z \vert ^{2n+2}}+ \frac{\sum_{k=0}^{n}(\overline{\xi_{k}-z_{k}})\bar {e}_{k}}{ \vert \xi-z \vert ^{2n+2}} \biggr)\,d\bar{\xi}\wedge d{\xi} \end{aligned}$$
is called T operator with B-M kernel.

Theorem 3.1

Let \(\Omega\subset C^{n+1}\) be a bounded domain, \(\varphi\in L^{p}(\overline{\Omega},\mathrm{Cl}_{0,n}(C))\), \(p>n+1\), then T is bounded on \(L^{p}(\Omega)\), and
$$\begin{aligned} \Vert T\varphi \Vert _{\Omega,p}\leq M_{1} \Vert \varphi \Vert _{\Omega,p}. \end{aligned}$$
(1)

Proof

Choose \(q>1\) such that \(\frac{1}{p}+\frac{1}{q}=1\), when \(p>2(n+1)\), we have \(1< q<\frac{2(n+1)}{2n+1}\), using Hölder’s inequality, we have
$$\begin{aligned} & \bigl\vert T\varphi(z) \bigr\vert \\ &\quad= \frac{1}{2^{n+1}\omega_{2n+2}} \biggl\vert \int_{\Omega}\varphi(\xi ) \biggl(\frac{\sum_{k=0}^{n}(\xi_{k}-z_{k})\bar{e}_{k}}{ \vert \xi-z \vert ^{2n+2}}+ \frac{\sum_{k=0}^{n}(\overline{\xi_{k}-z_{k}})\bar {e}_{k}}{ \vert \xi-z \vert ^{2n+2}} \biggr)\,d\bar{\xi}\wedge d{\xi} \biggr\vert \\ &\quad\leq\frac{K_{1}}{\omega_{2n+2}} \biggl( \int_{\Omega} \biggl\vert \varphi (\xi)\frac{\sum_{k=0}^{n}(\xi_{k}-z_{k})\bar{e}_{k}}{ \vert \xi -z \vert ^{2n+2}} \biggr\vert \,dV_{\xi} + \int_{\Omega} \biggl\vert \varphi(\xi)\frac{\sum_{k=0}^{n}(\overline{\xi _{k}-z_{k}})\bar{e}_{k}}{ \vert \xi-z \vert ^{2n+2}} \biggr\vert \,dV_{\xi} \biggr) \\ &\quad\leq K_{2} \biggl( \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \frac{1}{ \vert \xi -z \vert ^{2n+1}}\,dV_{\xi}+ \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \frac{1}{ \vert \xi-z \vert ^{2n+1}} \,dV_{\xi} \biggr) \\ &\quad \leq K_{3} \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \frac{1}{ \vert \xi-z \vert ^{2n+1}}\,dV_{\xi} \\ &\quad\leq K_{4} \Vert \varphi \Vert _{\Omega,p} \biggl( \int_{\Omega} \vert \xi -z \vert ^{-(2n+1)q}\,dV_{\xi} \biggr)^{\frac{1}{q}}. \end{aligned}$$
Because \(1< q<\frac{2(n+1)}{2n+1}\), we have \((2n+1)q<2(n+1)\). Using Lemma 2.2, for \(\forall z\in\Omega\), we have
$$\int_{\Omega} \vert \xi-z \vert ^{-(2n+1)q}\,dV_{\xi} \leq K_{5}. $$
So we have
$$\bigl\vert T\varphi(z) \bigr\vert \leq K_{4}K_{5} \Vert \varphi \Vert _{\Omega,p}. $$
Hence,
$$\biggl( \int_{\Omega} \bigl\vert T\varphi(z) \bigr\vert ^{p}\,dV_{z} \biggr)^{\frac{1}{p}}\leq K_{4}K_{5} \biggl( \int_{\Omega} \Vert \varphi \Vert _{\Omega,p}^{p}\,dV_{z} \biggr)^{\frac{1}{p}}. $$
Let \(M_{1}=K_{4}K_{5} (\int_{\Omega}\,dV_{z} )^{\frac{1}{p}}\), we have
$$\bigl\Vert T\varphi(z) \bigr\Vert _{\Omega,p}\leq M_{1} \Vert \varphi \Vert _{\Omega,p}. $$
 □

Theorem 3.2

Let \(z=z_{0}e_{0}+z_{1}e_{1}+z_{2}e_{2}+\cdots+z_{n}e_{n}\), \(\xi=\xi _{0}e_{0}+\xi_{1}e_{1}+\xi_{2}e_{2}+\cdots+\xi_{n}e_{n}\in\mathrm{Cl}_{0,n}(C)\), \(z\neq0\), \(\xi\neq0\), and \(|z|\neq|\xi|\), n (≥2), m (≥0) be integers, then for any i, \(0\leq i\leq n\), we have
$$\begin{aligned} \biggl\vert \frac{z_{i}}{ \vert z \vert ^{m+2}}-\frac{\xi_{i}}{ \vert \xi \vert ^{m+2}} \biggr\vert \leq \frac{ \vert z-\xi \vert [P_{m}(z,\xi)+ \vert z \vert ^{\frac{m}{2}} \vert \xi \vert ^{\frac{m}{2}} ]}{ \vert z \vert ^{m+1} \vert \xi \vert ^{m+1}}, \end{aligned}$$
(2)
where
$$P_{m}(z,\xi)= \sum_{k=0}^{m} \vert z \vert ^{m-k} \vert \xi \vert ^{k}={ \frac{ \vert z \vert ^{m+1}- \vert \xi \vert ^{m+1}}{ \vert z \vert - \vert \xi \vert }}. $$

Proof

Suppose \(|z|\leq|\xi|\) and insert the term \(z_{i}|z|^{m+2}\) in the following formula, then we have
$$\begin{aligned} & \biggl\vert \frac{z_{i}}{ \vert z \vert ^{m+2}}-\frac{\xi_{i}}{ \vert \xi \vert ^{m+2}} \biggr\vert \\ &\quad= \biggl\vert \frac{z_{i} \vert \xi \vert ^{m+2}-\xi_{i} \vert z \vert ^{m+2}}{ \vert z \vert ^{m+2} \vert \xi \vert ^{m+2}} \biggr\vert \\ &\quad= \biggl\vert \frac{z_{i} \vert \xi \vert ^{m+2}-z_{i} \vert z \vert ^{m+2}+z_{i} \vert z \vert ^{m+2}-\xi _{i} \vert z \vert ^{m+2}}{ \vert z \vert ^{m+2} \vert \xi \vert ^{m+2}} \biggr\vert \\ &\quad\leq\frac{ |z_{i}|| |\xi|^{m+2}- |z|^{m+2}|+|z_{i}-\xi _{i}| |z|^{m+2}}{ |z|^{m+2} |\xi|^{m+2}} \\ &\quad\leq\frac{|z|||\xi|-|z||(|\xi|^{m+1}+|\xi |^{m}|z|+\cdots+|z|^{m+1})+|z-\xi||z||\xi||z|^{m}}{|z|^{m+2}|\xi |^{m+2}} \\ &\quad\leq \biggl\vert \frac{|z|||\xi|-|z||(|\xi|^{m+1}+|\xi |^{m}|z|+\cdots+|\xi||z|^{m})+|z-\xi||z||\xi||z|^{m}}{|z|^{m+2}|\xi |^{m+2}} \biggr\vert \\ &\quad\leq \biggl\vert \frac{|z-\xi| [(|\xi|^{m}+|\xi|^{m-1}|z|+\cdots +|z|^{m})+|z|^{m} ]}{|z|^{m+1}|\xi|^{m+1}} \biggr\vert \\ &\quad\leq \biggl\vert \frac{|z-\xi| [P_{m}(z,\xi)+|z|^{\frac{m}{2}}|\xi |^{\frac{m}{2}} ]}{|z|^{m+1}|\xi|^{m+1}} \biggr\vert . \end{aligned}$$

When \(|\xi|\leq|z|\), insert \(\xi_{i}|\xi|^{m+2}\) in the above formula, we can prove the above inequality in a similar way. □

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 \(\Omega\subset C^{n+1}\) be a bounded domain, \(\varphi\in L^{p}(\Omega)\), \(p>2(n+1)\), then for any \(z_{1},z_{2}\in\Omega\), we have
$$\begin{aligned} \bigl\vert (T\varphi) (z_{1})-(T\varphi) (z_{2}) \bigr\vert \leq M_{2} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{\alpha}, \end{aligned}$$
(3)
and is Hölder continuous on Ω, where \(\alpha= 1-\frac{2(n+1)}{p}\).

Proof

Case 1. When \(|z_{1}-z_{2}|\geq1\), using Theorem 3.2 we have
$$\begin{aligned} \bigl\vert T\varphi(z_{1})-T\varphi(z_{2}) \bigr\vert \leq& 2M_{1} \Vert \varphi \Vert _{\Omega,p} \\ \leq& 2M_{1} \Vert \varphi \Vert _{\Omega,p} \frac {1}{ \vert z_{1}-z_{2} \vert ^{\alpha}} \vert z_{1}-z_{2} \vert ^{\alpha} \\ \leq& M_{2} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{\alpha}. \end{aligned}$$
Case 2. When \(|z_{1}-z_{2}|<1\), we have
$$\begin{aligned} & \bigl\vert T\varphi(z_{1})-T\varphi(z_{2}) \bigr\vert \\ &\quad\leq\frac{1}{\omega_{2n+2}2^{n+1}} \int_{\Omega} \bigl\vert \varphi(\xi ) \bigr\vert \biggl\vert \frac{\sum_{k=0}^{n}(\xi_{k}-z_{1k})\bar{e}_{k}}{ \vert \xi -z_{1} \vert ^{2n+2}} -\frac{\sum_{k=0}^{n}({\xi_{k}-z_{2k}})\bar{e}_{k}}{ \vert \xi -z_{2} \vert ^{2n+2}} \biggr\vert \vert d\bar{\xi}\wedge d{ \xi} \vert \\ &\qquad{}+ \frac{1}{\omega_{2n+2}2^{n+1}} \int_{\Omega} \bigl\vert \varphi(\xi ) \bigr\vert \biggl\vert \frac{\sum_{k=0}^{n}(\overline{\xi_{k}-z_{1k}})\bar{e}_{k}}{ \vert \xi -z_{1} \vert ^{2n+2}} -\frac{\sum_{k=0}^{n}(\overline{\xi_{k}-z_{2k}})\bar {e}_{k}}{ \vert \xi-z_{2} \vert ^{2n+2}} \biggr\vert \vert d\bar{\xi}\wedge d{ \xi} \vert \\ &\quad\leq\frac{1}{\omega_{2n+2}}\sum_{k=0}^{n} \int_{\Omega} \bigl\vert \varphi (\xi) \bigr\vert \biggl\vert \frac{(\xi_{k}-z_{1k})}{ \vert \xi-z_{1} \vert ^{2n+2}} -\frac{({\xi _{k}-z_{2k}})}{ \vert \xi-z_{2} \vert ^{2n+2}} \biggr\vert \,dV_{\xi} \\ &\qquad{}+\frac{1}{\omega_{2n+2}}\sum_{k=0}^{n} \int_{\Omega} \bigl\vert \varphi (\xi) \bigr\vert \biggl\vert \frac{(\overline{\xi_{k}-z_{1k}})}{ \vert \xi-z_{1} \vert ^{2n+2}} -\frac{(\overline{\xi_{k}-z_{2k}})}{ \vert \xi-z_{2} \vert ^{2n+2}} \biggr\vert \,dV_{\xi} \\ &\qquad=\frac{2}{\omega_{2n+2}}\sum_{k=0}^{n} \int_{\Omega} \bigl\vert \varphi(\xi ) \bigr\vert \biggl\vert \frac{(\xi_{k}-z_{1k})}{ \vert \xi-z_{1} \vert ^{2n+2}} -\frac{({\xi _{k}-z_{2k}})}{ \vert \xi-z_{2} \vert ^{2n+2}} \biggr\vert \,dV_{\xi}. \end{aligned}$$
Let
$$I= \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \biggl\vert \frac{(\xi_{k}-z_{1k})}{ \vert \xi -z_{1} \vert ^{2n+2}} -\frac{({\xi_{k}-z_{2k}})}{ \vert \xi-z_{2} \vert ^{2n+2}} \biggr\vert \,dV_{\xi}. $$
According to Theorem 3.2, we can get
$$\begin{aligned} I \leq& \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \biggl\vert \frac{ \vert z_{1}-z_{2} \vert [P_{2n}(\xi-z_{1},\xi-z_{2})+ \vert \xi-z_{1} \vert ^{n} \vert \xi-z_{2} \vert ^{n} ]}{ \vert \xi -z_{1} \vert ^{2n+1} \vert \xi-z_{2} \vert ^{2n+1}} \biggr\vert \,dV_{\xi} \\ =& \vert z_{1}-z_{2} \vert \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \biggl\vert \frac{P_{2n}(\xi -z_{1},\xi-z_{2})}{ \vert \xi-z_{1} \vert ^{2n+1} \vert \xi-z_{2} \vert ^{2n+1}} \biggr\vert \,dV_{\xi } \\ &{}+ \vert z_{1}-z_{2} \vert \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \frac{ \vert \xi-z_{1} \vert ^{n} \vert \xi -z_{2} \vert ^{n}}{ \vert \xi-z_{1} \vert ^{2n+1} \vert \xi-z_{2} \vert ^{2n+1}}\,dV_{\xi} \\ =&I_{1}+I_{2}. \end{aligned}$$
For \(I_{1}\), we have
$$\begin{aligned} I_{1} =& \vert z_{1}-z_{2} \vert \int_{\Omega}\sum_{k=0}^{2n} \vert \xi-z_{1} \vert ^{-(k+1)}|\xi-z_{2})|^{-(2n+1-k)}| \varphi(\xi)|dV_{\xi} \\ =& \vert z_{1}-z_{2} \vert \sum _{k=0}^{2n} \int_{\Omega} \vert \xi-z_{1} \vert ^{-(k+1)}| \xi-z_{2})|^{-(2n+1-k)}|\varphi(\xi)|dV_{\xi}. \end{aligned}$$
Using Hölder’s inequality we have
$$I_{1}\leq \vert z_{1}-z_{2} \vert \Vert \varphi \Vert _{\Omega,p} \sum_{k=0}^{2n} \biggl( \int_{\Omega} \vert \xi-z_{1} \vert ^{-(k+1)q} \vert \xi -z_{2} \vert ^{-(2n+1-k)q} \,dV_{\xi} \biggr)^{\frac{1}{q}}. $$
Because \(p>2n+2\), \(\frac{1}{p}+\frac{1}{q}=1\), we can get
$$1< q< \frac{2n+2}{2n+1}. $$
So
$$2n+1< (2n+1)q< 2n+2, $$
\(0\leq k\leq2n\), we get
$$\begin{gathered} (k+1)q\leq2n+2, \\ (2n+1-k)q\leq2n+2, \end{gathered} $$
and
$$(k+1)q+(2n+1-k)q>2n+2. $$
By Hadamard’s lemma, we have
$$\begin{aligned} & \int_{\Omega} \vert \xi-z_{1} \vert ^{-(k+1)q}| \xi-z_{2})|^{-(2n+1-k)q}\,dV_{\xi} \\ &\quad\leq K_{6} \vert z_{1}-z_{2} \vert ^{(2n+2)-(2n+1-k)q-(k+1)q} \\ &\quad=K_{6} \vert z_{1}-z_{2} \vert ^{(2n+2)-(2n+2)q}. \end{aligned}$$
So we have
$$\begin{aligned} I_{1} \leq& (2n+1)K_{6} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1+\frac{2n+2}{q}-(2n+2)} \\ \leq&(2n+1)K_{6} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1-\frac{2n+2}{p}}. \end{aligned}$$
As to \(I_{2}\), using Hölder’s inequality we have
$$\begin{aligned} I_{2} =& \vert z_{1}-z_{2} \vert \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert \frac{ \vert \xi -z_{1} \vert ^{n} \vert \xi-z_{2} \vert ^{n}}{ \vert \xi-z_{1} \vert ^{2n+1} \vert \xi-z_{2} \vert ^{2n+1}}\,dV_{\xi } \\ \leq& \vert z_{1}-z_{2} \vert \Vert \varphi \Vert _{\Omega,p} \biggl( \int_{\Omega } \vert \xi-z_{1} \vert ^{-(n+1)q} \vert \xi-z_{2} \vert ^{-(n+1)q} \,dV_{\xi} \biggr)^{\frac{1}{q}}. \end{aligned}$$
Since \(p>2n+2\) and \(\frac{1}{p}+\frac{1}{q}=1\), we get
$$1< q< \frac{2n+2}{2n+1}. $$
So
$$\begin{gathered} (n+1)q\leq2n+2, \\ (2n+2)q\geq2n+2. \end{gathered} $$
From Hadamard’s lemma, we get
$$\begin{aligned} & \int_{\Omega} \vert \xi-z_{1} \vert ^{-(n+1)q} \vert \xi-z_{2} \vert ^{-(n+1)q} \,dV_{\xi} \\ &\quad\leq K_{7} \vert z_{1}-z_{2} \vert ^{(2n+2)-(2n+2)q}. \end{aligned}$$
So
$$\begin{aligned} I_{2} \leq& K_{7} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1+\frac{2n+2}{q}-(2n+2)} \\ \leq& K_{7} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1-\frac{2n+2}{p}}. \end{aligned}$$
Hence
$$\begin{aligned} I =&I_{1}+I_{2} \\ \leq& (2n+1) (K_{6}+K_{7}) \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1-\frac{2n+2}{p}} \\ =&K_{8} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1-\frac {2n+2}{p}}. \end{aligned}$$
Using Hölder’s inequality, we obtain
$$\begin{aligned} \bigl\vert T\varphi(z_{1})-T\varphi(z_{2}) \bigr\vert \leq&\frac{2}{\omega_{2n+2}} K_{8} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1-\frac{2n+2}{p}} \\ \leq&K_{9} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{1-\frac{2n+2}{p}} \\ \leq&M_{2} \Vert \varphi \Vert _{\Omega,p} \vert z_{1}-z_{2} \vert ^{\alpha}. \end{aligned}$$
 □

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.

Theorem 3.4

Let \(\Omega\subset C^{n+1}\) be a bounded domain, \(\varphi\in L^{p}(\Omega)\), \(1< p<2n+2\), γ is an arbitrary constant which satisfies \(1<\gamma<\frac{(2n+2)p}{(2n+2)-p}\), then is γ-integrable on Ω, that is, \(T\varphi \in L^{\gamma}(\Omega)\), and the following inequality
$$\begin{aligned} \Vert T\varphi \Vert _{\Omega,\gamma}\leq M_{3} \Vert \varphi \Vert _{\Omega,p} \end{aligned}$$
(4)
is true.

Proof

For convenience, we introduce the notation b, suppose \(b=\frac {1}{\gamma}-\frac{1}{p}+\frac{1}{2n+2}\), then from\(1<\gamma<\frac{(2n+2)p}{(2n+2)-p}\) we know \(b>0\). Here are two cases to prove that is γ-integrable on Ω.

Case 1. When \(p<\gamma<\frac{(2n+2)p}{(2n+2)-p}\), \(0<\frac{p}{\gamma }<1\), thus \(0< p(\frac{1}{p}-\frac{1}{\gamma})=1-\frac{p}{\gamma}<1\), again
$$\frac{p}{\gamma}+p\biggl(\frac{1}{p}-\frac{1}{\gamma}\biggr)=1. $$
Choose \(q>0\) such that \(\frac{1}{p}+\frac{1}{q}=1\), then we have
$$\begin{aligned} &(2n+2) \biggl(\frac{b}{2}-\frac{1}{\gamma}\biggr)+(2n+2) \biggl( \frac{b}{2}-\frac{1}{q}\biggr) \\ &\quad=(2n+2) \biggl(b-\frac{1}{\gamma}-\frac{1}{q}\biggr) \\ &\quad=(2n+2) \biggl(\frac{1}{\gamma}-\frac{1}{p}+\frac{1}{2n+2}- \frac {1}{\gamma}-\frac{1}{q}\biggr) \\ &\quad=(2n+2) \biggl(-1+\frac{1}{2n+2}\biggr) \\ &\quad=-(2n+1). \end{aligned}$$
Therefore,
$$\begin{aligned} & \bigl\vert T\varphi(z) \bigr\vert \\ &\quad= \frac{1}{2^{n+1}\omega_{2n+2}} \biggl\vert \int_{\Omega}\varphi(\xi ) \biggl(\frac{\sum_{k=0}^{n}(\xi_{k}-z_{k})\bar{e}_{k}}{ \vert \xi-z \vert ^{2n+2}}+ \frac{\sum_{k=0}^{n}(\overline{\xi_{k}-z_{k}})\bar{e}_{k}}{ \vert \xi -z \vert ^{2n+2}} \biggr)\,d\bar{\xi}\wedge d{\xi} \biggr\vert \\ &\quad\leq\frac{K_{10}}{\omega_{2n+2}} \biggl( \int_{\Omega} \bigl\vert \varphi (\xi) \bigr\vert \frac{1}{ \vert \xi-z \vert ^{2n+1}}\,dV_{\xi} + \int_{\Omega} \bigl\vert \varphi (\xi) \bigr\vert \frac{1}{ \vert \xi-z \vert ^{2n+1}}\,dV_{\xi} \biggr) \\ &\quad= \frac{2K_{10}}{\omega_{2n+2}} \int_{\Omega} \bigl\vert \varphi(\xi ) \bigr\vert \frac{1}{ \vert \xi-z \vert ^{2n+1}}\,dV_{\xi} \\ &\quad= \frac{2K_{10}}{\omega_{2n+2}} \int_{\Omega} \bigl\vert \varphi(\xi ) \bigr\vert ^{\frac{p}{\gamma}} \vert \xi-z \vert ^{(2n+2)(\frac{b}{2}-\frac{1}{\gamma})} \bigl\vert \varphi(\xi) \bigr\vert ^{p(\frac{1}{p}-\frac{1}{\gamma})} \vert \xi -z \vert ^{(2n+2)(\frac{b}{2}-\frac{1}{q})}\,dV_{\xi}. \end{aligned}$$
Because \(1< p<\gamma\), \(\frac{1}{\gamma}+(\frac{1}{p}-\frac{1}{\gamma })+\frac{1}{q}=1\), using Hölder’s inequality we get
$$\begin{aligned} & \bigl\vert T\varphi(z) \bigr\vert \\ &\quad\leq\frac{2K_{10}}{\omega_{2n+2}} \biggl( \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert ^{p} \vert \xi-z \vert ^{(2n+2)(\frac{\gamma b}{2}-1)}\,dV_{\xi } \biggr)^{\frac{1}{\gamma}} \biggl( \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert ^{p}\,dV_{\xi} \biggr)^{\frac {1}{p}-\frac{1}{\gamma}} \\ & \qquad{} \cdot \biggl( \int_{\Omega} \vert \xi-z \vert ^{(2n+2)(\frac {qb}{2}-1)}\,dV_{\xi} \biggr)^{\frac{1}{q}} \\ &\quad= \frac{2K_{10}}{\omega_{2n+2}} \biggl( \int_{\Omega} \bigl\vert \varphi (\xi) \bigr\vert ^{p} \vert \xi-z \vert ^{(2n+2)(\frac{\gamma b}{2}-1)}\,dV_{\xi} \biggr)^{\frac{1}{\gamma}} \Vert \varphi \Vert _{\Omega,p}^{1-\frac{p}{\gamma }} \\ & \qquad{} \cdot \biggl( \int_{\Omega} \vert \xi-z \vert ^{(2n+2)(\frac {qb}{2}-1)}\,dV_{\xi} \biggr)^{\frac{1}{q}}. \end{aligned}$$
Because \(b>0\), we have
$$\begin{gathered} (2n+2) \biggl(1-\frac{\gamma b}{2}\biggr)< 2n+2, \\ (2n+2) \biggl(1-\frac{qb}{2}\biggr)< 2n+2. \end{gathered} $$
From Lemma 2.2 we can know that two integrals are meaningful, we assume that \(K_{11}= \sup_{\xi\in\Omega} \int_{\Omega}|\xi -z|^{(2n+2)(\frac{qb}{2}-1)}\,dV_{\xi}\).
Therefore we have
$$\bigl\vert T\varphi(z) \bigr\vert ^{\gamma}\leq \biggl( \frac{2}{\omega_{2n+2}} \biggr)^{\gamma }K_{11}^{\frac{\gamma}{q}} \Vert \varphi \Vert _{\Omega,p}^{\gamma-p} \biggl( \int_{\Omega} \bigl\vert \varphi(\xi) \bigr\vert ^{p} \vert \xi-z \vert ^{(2n+2)(\frac{\gamma b}{2}-1)}\,dV_{\xi } \biggr). $$
Let \(K_{12}= \sup_{\xi\in\Omega} \int_{\Omega}|\xi -z|^{(2n+2)(\frac{\gamma b}{2}-1)}\,dV_{z}\), so we have
$$\bigl\vert T\varphi(z) \bigr\vert ^{\gamma}\leq K_{12} \Vert \varphi \Vert _{\Omega,p}^{\gamma -p} \Vert \varphi \Vert _{\Omega,p}^{p}=K_{13} \Vert \varphi \Vert _{\Omega,p}^{\gamma}, $$
where \(K_{13}= (\frac{2}{\omega_{2n+2}} )^{\gamma}K_{11}^{\frac {\gamma}{q}}K_{12}\).
Hence, we get
$$\Vert T\varphi \Vert _{\Omega,\gamma}= \biggl( \int_{\Omega} \bigl\vert T\varphi (z) \bigr\vert ^{\gamma}\,dV_{\xi} \biggr)^{\frac{1}{\gamma}} \leq K_{13}^{\gamma} \Vert \varphi \Vert _{\Omega,p}=K_{14} \Vert \varphi \Vert _{\Omega,p}, $$
where \(K_{14}= K_{13}^{\gamma}\).

(2) When \(p\geq\gamma>1\), choose m such that \(0<\frac{(2n+2)\gamma }{(2n+2)+\gamma}<m<\gamma\), and m is an arbitrary positive constant satisfying \(m<\gamma<\frac {(2n+2)m}{(2n+2)+m}\). Because \(\varphi\in L^{p}(\Omega)\), \(m< p\), we have \(\varphi\in L^{m}(\Omega)\).

Choose \(\frac{1}{p}+\frac{1}{q}=\frac{1}{m}\). Therefore, from the proof process of (1) and Lemma 2.5, we get
$$\begin{aligned} & \Vert T\varphi \Vert _{\Omega,p} \\ &\quad\leq K_{15} \Vert \varphi \Vert _{\Omega,m} \\ &\quad= K_{15} \biggl[ \int_{\Omega} \bigl\vert \varphi(\xi)\cdot1 \bigr\vert ^{m} \biggr]^{\frac{1}{m}} \\ &\quad\leq K_{15} \biggl[ \int_{\Omega} \bigl\vert \varphi(\xi)\cdot1 \bigr\vert ^{p} \biggr]^{\frac{1}{p}} \biggl[ \int_{\Omega} \bigl\vert \varphi(\xi)\cdot1 \bigr\vert ^{q} \biggr]^{\frac{1}{q}} \\ &\quad\leq K_{15}V_{\Omega}^{\frac{1}{q}} \Vert \varphi \Vert _{\Omega,p} \\ &\quad\leq K_{16} \Vert \varphi \Vert _{\Omega,p}. \end{aligned}$$
Therefore is γ integrable on Ω. If we choose \(M_{3}=\max\{K_{14},K_{16}\}\), then
$$\Vert T\varphi \Vert _{\Omega,\gamma}\leq M_{3} \Vert \varphi \Vert _{\Omega,p}. $$
 □

Declarations

Funding

This work was supported by the National Science Foundation of China (Nos. 11571089 and 11871191 ), the Natural Science Foundation of Hebei Province (No. A2016205218, No. CXZZBS2017085, No. A2015205012), and the Key Foundation of Hebei Normal University (No. L2018Z01).

Authors’ contributions

ZFL has presented the main purpose of the article. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Science, Hebei University of Science and Technology, Shijiazhuang, P.R. China
(2)
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang, P.R. China
(3)
Hebei GEO University, Shijiazhuang, P.R. China

References

  1. Brack, F., Delanghe, R., Sommen, F.: Clifford Analysis. Pitman, Boston (1982) MATHGoogle Scholar
  2. Eriksson, S.L., Leutwiler, H.: Hypermonogenic function. In: Clifford Algebras and Their Applications in Mathematical Physics, pp. 287–302. Birkhäuser, Boston (2000) View ArticleGoogle Scholar
  3. Eriksson, S.L.: Integral formulas for hypermonogenic functions. Bull. Belg. Math. Soc. Simon Stevin 11, 705–718 (2004) MathSciNetMATHGoogle Scholar
  4. Eriksson, S.L., Leutwiler, H.: An improved Cauchy formula for hypermonogenic functions. Adv. Appl. Clifford Algebras 19, 269–282 (2009) MathSciNetView ArticleMATHGoogle Scholar
  5. Eriksson, S.L., Leutwiler, H.: Hypermonogenic functions and their Cauchy-type theorems. In: Advances in Analysis and Geometry, pp. 97–112 (2004) View ArticleGoogle Scholar
  6. Huang, S., Qiao, Y.Y., Wen, G.C.: Real and Complex Clifford Analysis. Springer, New York (2006) MATHGoogle Scholar
  7. Qiao, Y.Y., Ryan, J.: Orthogonal projections on hyperbolic space. In: Harmonic Analysis, Signal Processing, and Complexity. Progress in Mathematics, vol. 238, pp. 111–120. Birkhaäuser, Boston (2005) View ArticleGoogle Scholar
  8. Qiao, Y.Y., Bernstein, S., Eriksson, S.L.: Function theory for Laplace and Dirac–Hodge operators on hyperbolic space. J. Anal. Math. 98, 43–63 (2006) MathSciNetView ArticleMATHGoogle Scholar
  9. Li, Z.F., Yang, H.J., Qiao, Y.Y., Guo, B.C.: Some properties of T-operator with bihypermonogenic kernel in Clifford analysis. Complex Var. Elliptic Equ. 62, 938–956 (2017) MathSciNetView ArticleMATHGoogle Scholar
  10. Xie, Y.H., Zhang, X.F., Tang, X.M.: Some properties of k-hypergenic functions in Clifford analysis. Complex Var. Elliptic Equ. 61, 1614–1626 (2016) MathSciNetView ArticleMATHGoogle Scholar
  11. Xie, Y.H.: Boundary properties of hypergenic-Cauchy type integrals in Clifford analysis. Complex Var. Elliptic Equ. 59, 599–615 (2014) MathSciNetView ArticleMATHGoogle Scholar
  12. Xie, Y.H., Yang, H.J., Qiao, Y.Y.: Complex k-hypermonogenic functions in complex Clifford analysis. Complex Var. Elliptic Equ. 58, 1467–1479 (2013) MathSciNetView ArticleMATHGoogle Scholar
  13. Yang, H.J., Zhao, X.H.: The fixed point and Mann iterative of a kind of higher order singular Teodorescu operator. Complex Var. Elliptic Equ. 60, 1658–1667 (2015) MathSciNetView ArticleMATHGoogle Scholar
  14. Yang, H.J., Qiao, Y.Y., Xie, Y.H., Wang, L.P.: Cauchy integral formula for k-monogenic function with α-weight applied Clifford algebra. Adv. Appl. Clifford Algebras 28, 1–14 (2018) MathSciNetView ArticleGoogle Scholar
  15. Yang, H.J., Qiao, Y.Y., Huang, S.: Some properties of Cauchy-type singular integrals in Clifford analysis. J. Math. Res. Appl. 32, 189–200 (2012) MathSciNetMATHGoogle Scholar
  16. Ryan, J.: Complexied Clifford analysis. Complex Var. Theory Appl. 1, 119–149 (1982) Google Scholar
  17. Ryan, J.: Singularities and Laurent expansions in complex Clifford analysis. Appl. Anal. 16, 33–49 (1983) MathSciNetView ArticleMATHGoogle Scholar
  18. Ryan, J.: Iterated Dirac operators in \(C^{n}\). Z. Anal. Anwend. 9, 385–401 (1990) View ArticleMATHGoogle Scholar
  19. Ryan, J.: Intrinsic Dirac operators in \(C^{n}\). Adv. Math. 118, 93–133 (1996) View ArticleGoogle Scholar
  20. Ku, M., Du, J.Y., Wang, D.S.: Some properties of holomorphic Cliffordian functions in complex Clifford analysis. Acta Math. Sci. 30, 747–768 (2010) MathSciNetView ArticleMATHGoogle Scholar
  21. Ku, M., Du, J.Y., Wang, D.S.: On generalization of Martinelli–Bochner integral formula using Clifford analysis. Adv. Appl. Clifford Algebras 20, 351–366 (2010) MathSciNetView ArticleMATHGoogle Scholar
  22. Gilbert, R.P., Buchanan, J.L.: First Order Elliptic Systems, a Function Theoretic Approach. Academic Press, New York (1983) MATHGoogle Scholar
  23. Vekua, I.N.: Generalized Analytic Functions. Pergamon, Oxford (1962) MATHGoogle Scholar

Copyright

© The Author(s) 2018

Advertisement