1 Introduction

The Clifford algebra \({\mathcal{A}}_{n}(R)\) was established by Clifford [1] in 1878, which is the extension of complex numbers, quaternions, and exterior algebras. It is an associative and incommutable algebra structure. And it possesses both theoretical and applicable value to many fields, such as quantum mechanics, quantum field theory [2], projective geometry, computer graphics [3–5], neural network theory [6, 7], and so on. Clifford analysis is an important branch of modern analysis, which studies functions defined on \(R^{n}\) with their values in Clifford algebra space \({\mathcal {A}}_{n}(R)\). Clifford analysis is an important tool of modern mathematics and physics. In addition, Clifford analysis is now a widely studied field [8]: function theory, harmonic analysis, potential theory, partial differential equations, differential geometry. Since 1970, on the basis of the Dirac operator, Brackx et al. [9–12] etc. put forward the regular function, which is an extension of the holomorphic function in higher dimensional space. Thus they lay a theoretical foundation for Clifford analysis. The k-regular function is a natural generalization of the regular functions. In 1976, Brackx [13] was first to introduce k-regular functions of the real quaternion and gave the Cauchy integral formula and Taylor expansion. In 1977, Delanghe and Brackx [14] studied k-regular functions which were defined on \(R^{n}\) with values in Clifford algebra space \({\mathcal{A}}_{n}(R)\) and obtained the corresponding Cauchy integral formula. Recently, many scholars such as Begehr et al. [15], Li et al. [16, 17], Zeng and Yang [18] etc. have studied some properties and corresponding boundary value problems of k-regular functions.

Teodorescu operator is the generalized solution of non-homogeneous Dirac equation. Therefore it plays a key role in studying the integral expression of non-homogeneous Dirac equation and many boundary value problems. In addition, it also have important applications in many subjects, such as physics, chemistry, engineering technology, etc. Due to its good properties and important applications, it has been a much studied and significant topic. Vekua [19] first discussed some properties of the Teodorescu operator detailedly, and Hile [20] studied some properties of the Teodorescu operator in \(R^{n}\). Then Gilbert et al. [21] and Meng [22] studied its many properties in high dimensional complex space. Gürlebeck and Sprössig [23], and Yang [24] discussed its properties and the corresponding boundary value problems in the real quaternion analysis. Wang and Gong [25] discussed the stabilities of the singular integral operators when the boundary curve of integral domain is perturbed. Recently, Brackx et al. [26] studied some properties of the Teodorescu operator which is related to the Hermitian regular functions. Wang et al. [27, 28] showed some properties of the Teodorescu operator and corresponding boundary value problems.

In this paper, we define a kind of generalized Teodorescu operator which is related to the k-regular functions in real Clifford analysis. First, we discuss the boundedness and Hölder continuity for the generalized Teodorescu operator in a nonempty open bounded connected domain in \(R^{n}\). Second, we discuss the stability and give the error estimate of the generalized Teodorescu operator when the boundary surface of the integral domain is perturbed. These results make the theory of Clifford analysis more perfect and also lay a theoretical foundation for studying the properties of singularity integral operator in Clifford analysis.

2 Preliminaries

Let \({e_{1},\ldots,e_{n}}\) be an orthogonal basis of the Euclidean space \(R^{n}\), and let \({\mathcal{A}}_{n}(R)\) be the Clifford algebra with basis \({\{e_{A}:e_{A}=e_{\alpha_{1}}\cdots e_{\alpha_{h}}\}}\), where \(A=\{\alpha_{1},\ldots, \alpha_{h}\} \subseteq\{1,\ldots,n\}\), \(1\leq\alpha_{1}<\alpha_{2}<\cdots<\alpha _{h}\leq n\) and \(e_{\emptyset}=e_{0}=1\). The associative and noncommutative multiplication of the basis in \({\mathcal{A}}_{n}(R)\) is governed by the rules:

$$\left \{ \textstyle\begin{array}{l} e_{i}^{2}=-1, \quad i=1,2,\ldots,n, \\ e_{i}e_{j}=-e_{j}e_{i}, \quad 1\leq i,j\leq n, i\neq j, \\ e_{0}e_{i}=e_{i}, \quad i=1,2,\ldots,n. \end{array}\displaystyle \right . $$

Hence the real Clifford algebra is composed of elements having the type \(a=\sum_{A}x_{A}e_{A}\), \(x_{A}\in R\). The norm for an element \(a\in{\mathcal{A}}_{n}(R)\) is taken to be \(|a|=\sqrt{\sum_{A}|x_{A}|^{2}}\) and satisfies \(|\bar{a}|=|a|\), \(|a+b|\leq{|a|+|b|}\), \(|ab|\leq2^{n}|a||b|\).

In addition, we suppose \(\Omega\subset R^{n}\) (\(n\geq2\)) is a bounded connected domain and the boundary ∂Ω is a differentiable, oriented, and compact Liapunov surface. The function f, which is defined in Ω with values in \({\mathcal{A}}_{n}(R)\), can be expressed as \(f(x)=\sum_{A}f_{A}(x)e_{A}\), where the \(f_{A}(x)\) are real-valued functions. In this paper, let \(f(x)\in{C^{(r)}(\Omega,{\mathcal {A}}_{n}(R))}=\{f\mid f:\Omega\rightarrow{\mathcal{A}}_{n}(R), f(x)=\sum_{A}f_{A}(x)e_{A}\}\), where \(f_{A}(x)\) has continuous r times differentials. The Dirac operator is defined as follows:

$$Df={\sum_{i=1}^{n}} {e_{i} \frac{\partial f}{\partial x_{i}}}=\sum_{i=1}^{n}{\sum _{A}} {e_{i}e_{A} \frac {\partial f_{A}}{\partial x_{i}}}, \qquad fD={\sum_{i=1}^{n}} {\frac{\partial f}{\partial x_{i}}e_{i}}={\sum_{i=1}^{n}} {\sum_{A}} {e_{A}e_{i} \frac{\partial f_{A}}{\partial x_{i}}}. $$

If \(Df(x)=0\) (\(f(x)D=0\)) (\(x\in\Omega\)), then f is called left(right) regular function. Furthermore, if \(D^{k}f(x)=0\) (\(f(x)D^{k}=0\)) (\(x\in\Omega \)), where \(r\geq k\), \(k< n\). Then f is called left (right) k-regular function. Usually left regular (k-regular) is called regular (k-regular) for short. And obviously, when \(k=1\), a k-regular function is a regular function. \(f(x)\in{L^{p}(\Omega)}\) means \(D^{j}{f(x)}\in{ L^{p}(\Omega )}\), \(j=0,1,\ldots,k-1\), and \(L_{p}[f(x),\Omega]={\sum_{j=0}^{k-1}}{L_{p}[D^{j}{f(x)},\Omega]}\). Denote \(B(\rho_{0})=\{\omega \mid \omega = \sum_{i=1}^{n}\omega_{i}(x)e_{i}, \omega_{i}\in {C^{1}(\partial\Omega)},\omega_{n}(x)\geq 0,\|\omega\|_{\partial\Omega}<\rho_{0}\}\), where \(\|\omega\| _{\partial\Omega}= \max_{x\in \partial\Omega}\sqrt{\sum_{i=1}^{n}\sqrt{\sum_{j=1}^{n}|\frac{\partial(\omega_{i}(x))}{\partial x_{j}}|^{2}}} +\max_{x\in \partial\Omega}|\omega(x)|\).

In the following, we define a kind of generalized singular Teodorescu operator and give some necessary lemmas. Then we discuss some properties of the generalized singular Teodorescu operator.

Definition 2.1

Let Ω be as stated above and \(f(x)\in{C^{(r)}(\Omega ,{\mathcal{A}}_{n}(R))}\), \(D^{j}{f(x)}\in{ L^{p}(\Omega )}\) (\(j=0,1,\ldots,k-1\)), we define a kind of generalized singular Teodorescu operator as follows:

$$\bigl(T_{\Omega}[f]\bigr) (y)=\sum_{j=0}^{k-1}(-1)^{j} \int _{\Omega}\frac{A_{j+1}}{\omega_{n}}\frac{D^{j}f(x)(\bar{x}-\bar {y})^{j+1}}{|x-y|^{n+\alpha}}\, dx, $$

where \(0<\alpha<1\) is a positive constant and \(\omega_{n}\) is the area of the unit sphere in \(R^{n}\), \(A_{j}\) is a constant as stated in [15], it is irrelevant to the x, y.

Remark 2.1

When \(\alpha=0\), \(n=2\), \(k=1\), the singular integral operator is a normal Teodorescu operator.

Remark 2.2

When \(y\in\Omega^{-}=R^{n}-\overline{\Omega}\), \(T[f]\) is a normal generalized integral. When \(y\in\overline{\Omega}\), \(T[f]\) is a kind of generalized singular integral.

Lemma 2.1

([2])

Let \(\Omega\subset R^{n}\) (\(n\geq2\)) be as stated above. When \(\mu< n\), for any \(y\in R^{n}\), we have \({\int_{\Omega }|x-y|^{-\mu}\, dx\leq M_{3}}\), where \(M_{3}\) is a constant number which only depends on μ, Ω.

Lemma 2.2

([16])

For any \(x,y_{1},y_{2}\in R^{n}\), when \(j\geq0\), we have

$$\biggl\vert \frac{(x-y_{1})^{j+1}}{|x-y_{1}|^{n}} -\frac{(x-y_{2})^{j+1}}{|x-y_{2}|^{n}}\biggr\vert \leq \Biggl[\sum _{i=1}^{n-1}\frac{(|x|+ |y_{2}|)^{j}}{|x-y_{1}|^{i}|x-y_{2}|^{n-i}}+\sum _{i=1}^{j}\frac{(|x|+|y_{2}|)^{j-i}}{|x-y_{1}|^{n-i}} \Biggr]|y_{1}-y_{2}|. $$

Remark 2.3

When \(j=0 \), the second part \(\sum_{i=1}^{j}\frac {(|x|+|y_{2}|)^{j-i}}{|x-y_{1}|^{n-i}}\) is vanishing.

Lemma 2.3

([29])

If \(\sigma_{1},\sigma_{2}>0\), \(0\leq\alpha\leq1\), then we have \(|\sigma_{1}^{\alpha}-\sigma_{2}^{\alpha}|\leq|\sigma_{1}-\sigma _{2}|^{\alpha}\).

Lemma 2.4

([21])

Suppose Ω is a bounded domain in \(R^{n}\) (\(n\geq2\)), and let \(\alpha'\), \(\beta'\) satisfy \(0<\alpha', \beta'<n\), \(\alpha'+\beta '>n\). Then, for all \(x_{1},x_{2}\in R^{n}\) and \(x_{1}\neq x_{2}\), we have

$$\int_{\Omega}|t-x_{1}|^{-\alpha'}|t-x_{2}|^{-\beta'} \, dt\leq M_{1}\bigl(\alpha',\beta' \bigr)|x_{1}-x_{2}|^{n-\alpha'-\beta'}. $$

Lemma 2.5

([25])

Let Ω be as stated above. Suppose the area of ∂Ω is \(A_{\partial\Omega}\) and \(\partial\Omega_{\omega}= \{t+\omega(t)\mid t\in\partial\Omega, \omega(t)\in B(\rho_{0})\}\). Suppose \(\Omega_{\omega}\) is the inside domain surrounded by \(\partial\Omega_{\omega}\) and \(\Omega_{\omega }^{-}=R^{n}-\Omega_{\omega}\), \(E_{1}=\Omega_{\omega}\cap {\Omega}^{-}\), \(E_{2}=\Omega_{\omega}^{-}\cap{\Omega}\). Then we have \(A_{E_{1}\cup E_{2}}\leq M_{2}\|\omega\|_{\partial\Omega}\).

3 Some properties of generalized Teodorescu operator

Theorem 3.1

Let \(\Omega\subset R^{n}\) (\(n\geq2\)) be as stated above, \(f(x)\in{C^{(r)}(\Omega,{\mathcal {A}}_{n}(R))}\), \(D^{j}{f(x)}\in{ L^{p}(\Omega)}\) (\(j=0,1,\ldots,k-1\)), \(p>n/(1-\alpha)\). Then \(T_{\Omega}[f](y)\) is uniformly bounded on \(R^{n}\), and we have

$$\bigl\vert T_{\Omega}[f](y)\bigr\vert \leq M_{4}(n,p, \alpha,\Omega)L_{p}[f,\Omega]. $$

Proof

By the Hölder inequality, we have

$$\begin{aligned} \bigl\vert T_{\Omega}[f](y)\bigr\vert =&\Biggl\vert {\sum _{j=0}^{k-1}(-1)^{j} \int_{\Omega}\frac{A_{j+1}}{\omega_{n}}\frac {D^{j}f(x)(\bar{x}-\bar{y})^{j+1}}{|x-y|^{n+\alpha}}\, dx}\Biggr\vert \\ \leq&J_{1} {\sum_{j=0}^{k-1} \biggl[ \int_{\Omega }\bigl\vert D^{j}f(x)\bigr\vert ^{p}|dx| \biggr]^{\frac{1}{p}} \biggl[ \int_{\Omega}\frac {|dx|}{|x-y|^{(n+\alpha-j-1)q}} \biggr]^{\frac{1}{q}}} \\ \leq&J_{2}L_{p}[f,\Omega] {\sum _{j=0}^{k-1} \biggl[ \int _{\Omega}\frac{|dx|}{|x-y|^{(n+\alpha-j-1)q}} \biggr]^{\frac{1}{q}}}. \end{aligned}$$

Again from \(p>n/(1-\alpha)\), \(1/p+1/q=1\), we have \(1< q< n/(n+\alpha-1)\). Thus

$$(n+\alpha-j-1)q\leq(n+\alpha-1)q< n, \quad j=0,1,\ldots,k-1. $$

Hence for all \({j=0,1,\ldots,k-1}\), the integral \(\int _{\Omega}|x-y|^{-(n+\alpha-j-1)q}|dx|\) is convergent. Thus it is bounded. And from Lemma 2.1, we can know its boundary is independent of y. Thus for all \(y\in R^{n}\), we have

$$\bigl\vert T_{\Omega}[f](y)\bigr\vert \leq M_{4}(n,p, \alpha,\Omega)L_{p}[f,\Omega]. $$

 □

Theorem 3.2

Let Ω be as stated above, \(f(x)\in{C^{(r)}(\Omega,{\mathcal {A}}_{n}(R))}\), \(D^{j}{f(x)}\in{ L^{p}(\Omega)}\) (\(j=0,1,\ldots, k-1\)), \(p>n/(1-\alpha)\). Then we have the following results:

  1. (1)

    If \(1/2\leq\alpha<1\), then, for any \(y_{1},y_{2}\in\Omega \), we can obtain

    $$\bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \leq M_{5}(n,p,\alpha ,\Omega)L_{p}[f, \Omega]|y_{1}-y_{2}|^{\beta}, $$

    where \(\beta=1-\alpha-n/p\), \(0<\beta<1\).

  2. (2)

    If \(0<\alpha<1/2\), let \(p_{1}\) be a constant and satisfies \(p_{1}< p\), \(n/(1-\alpha)< p_{1}< n/(1-2\alpha)\), then, for any \(y_{1},y_{2}\in\Omega\), we have

    $$\bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \leq M_{6}(n,p,\alpha ,\Omega)L_{p_{1}}[f, \Omega]|y_{1}-y_{2}|^{\gamma}, $$

    where \(\gamma=1-\alpha-n/p_{1}\), \(0<\gamma<1\).

Proof

(1) From Definition 2.1, we get

$$\begin{aligned}& \bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \\& \quad = \Biggl\vert \sum_{j=0}^{k-1}(-1)^{j} \int_{\Omega}\frac {A_{j+1}}{\omega_{n}}\frac{D^{j}f(x)(\bar{x}-\bar {y}_{1})^{j+1}}{|x-y_{1}|^{n+\alpha}}\, dx \\& \qquad {} - \sum_{j=0}^{k-1}(-1)^{j} \int_{\Omega}\frac {A_{j+1}}{\omega_{n}}\frac{D^{j}f(x)(\bar{x}-\bar {y}_{2})^{j+1}}{|x-y_{2}|^{n+\alpha}}\, dx\Biggr\vert \\& \quad \leq J_{3} \sum_{j=0}^{k-1} \int_{\Omega }\bigl\vert D^{j}f(x)\bigr\vert \biggl\vert \frac{(\bar{x}-\bar {y}_{1})^{j+1}}{|x-y_{1}|^{n+\alpha}}-\frac{(\bar{x}-\bar {y}_{2})^{j+1}}{|x-y_{2}|^{n+\alpha}}\biggr\vert |dx|. \end{aligned}$$
(3.1)

Again, from Lemma 2.2, Lemma 2.3, and Ω being a bounded domain, we obtain

$$\begin{aligned}& \biggl\vert \frac{(\bar{x}-\bar {y}_{1})^{j+1}}{|x-y_{1}|^{n+\alpha}}-\frac{(\bar{x}-\bar {y}_{2})^{j+1}}{|x-y_{2}|^{n+\alpha}}\biggr\vert \\& \quad \leq \frac{1}{|x-y_{1}|^{\alpha}} \Biggl[\sum_{i=1}^{n-1} \frac {(|x|+|y_{2}|)^{j}}{|x-y_{1}|^{i}|x-y_{2}|^{n-i}}+\sum_{i=1}^{j} \frac {(|x|+|y_{2}|)^{j-i}}{|x-y_{1}|^{n-i}} \Biggr]|y_{1}-y_{2}| \\& \qquad {}+ \frac{1}{|x-y_{2}|^{n-j-1}}\biggl\vert \frac {1}{|x-y_{1}|}-\frac{1}{|x-y_{2}|} \biggr\vert ^{\alpha} \\& \quad \leq \frac{1}{|x-y_{1}|^{\alpha}}\sum_{i=1}^{n-1} \frac {(|x|+|y_{2}|)^{j}}{|x-y_{1}|^{i}|x-y_{2}|^{n-i}}|y_{1}-y_{2}| \\& \qquad {}+ \frac{1}{|x-y_{1}|^{\alpha}}\sum_{i=1}^{j} \frac{(|x|+|y_{2}|)^{j-i}}{|x-y_{1}|^{n-i}}|y_{1}-y_{2}| + \frac{1}{|x-y_{2}|^{n-j-1}} \frac{|y_{1}- y_{2}|^{\alpha}}{|x-y_{1}|^{\alpha}|x-y_{2}|^{\alpha}} \\& \quad \leq J_{4} \frac{1}{|x-y_{1}|^{\alpha}}\sum _{i=1}^{n-1}\frac{1}{|x-y_{1}|^{i}|x-y_{2}|^{n-i}}|y_{1}-y_{2}| \\& \qquad {}+ J_{5}\frac{1}{|x-y_{1}|^{\alpha}}\sum_{i=1}^{j} \frac{1}{|x-y_{1}|^{n-i}}|y_{1}-y_{2}|+ \frac {|y_{1}-y_{2}|^{\alpha}}{|x-y_{1}|^{\alpha}|x-y_{2}|^{n+\alpha-j-1}}. \end{aligned}$$
(3.2)

Thus, from the above, we get

$$\begin{aligned}& \bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \\& \quad \leq J_{6} {\sum_{j=0}^{k-1} \sum_{i=1}^{n-1} \int _{\Omega}\frac{1}{|x-y_{1}|^{i+\alpha }|x-y_{2}|^{n-i}}\bigl\vert D^{j}f(x) \bigr\vert |dx||y_{1}-y_{2}|} \\& \qquad {}+J_{7} {\sum_{j=1}^{k-1} \sum_{i=1}^{j} \int_{\Omega }\frac{1}{|x-y_{1}|^{n+\alpha-i}}\bigl\vert D^{j}f(x) \bigr\vert |dx||y_{1}-y_{2}|} \\& \qquad {}+J_{8} {\sum_{j=0}^{k-1} \int_{\Omega}\frac {1}{|x-y_{1}|^{\alpha}|x-y_{2}|^{n+\alpha -j-1}}\bigl\vert D^{j}f(x) \bigr\vert |dx||y_{1}-y_{2}|^{\alpha}} \\& \quad = I_{1}+I_{2}+I_{3}. \end{aligned}$$
(3.3)

In the following, first we evaluate \(I_{1}\). From the Hölder inequality, we have

$$\begin{aligned} I_{1} \leq&J_{9}L_{p}[f,\Omega] {\sum _{i=1}^{n-1} \biggl[ \int_{\Omega}|x-y_{1}|^{-(i+\alpha)q}|x-y_{2}|^{-(n-i)q}|dx| \biggr]^{\frac{1}{q}} \cdot|y_{1}-y_{2}|} \\ =&J_{9}L_{p}[f,\Omega]|y_{1}-y_{2}| {\sum_{i=1}^{n-1}{I_{1}}_{i}^{\frac{1}{q}}}, \end{aligned}$$
(3.4)

where \({I_{1}}_{i}=\int_{\Omega}|x-y_{1}|^{-(i+\alpha )q}|x-y_{2}|^{-(n-i)q}|dx|\), \(i=1,2,\ldots,n-1\).

In addition, when \(p>n/(1-\alpha)\), we have \(1< q< n/(n+\alpha-1)\). Thus let \(\alpha'=(i+\alpha)q\), \(\beta'=(n-i)q\), then

$$\begin{aligned}& 0< \alpha'=(i+\alpha)q< (i+\alpha)\frac{n}{n+\alpha-1}\leq \frac {(n-1+\alpha)n}{n+\alpha-1}=n, \\& 0< \beta'=(n-i)q< (n-i)\frac{n}{n+\alpha-1}\leq\frac {n(n-1)}{n+\alpha-1}< n. \end{aligned}$$

And \(\alpha'+\beta'=(n+\alpha)q>(n+\alpha)>n\).

Thus from Lemma 2.4, we have

$${I_{1}}_{i}\leq J_{10}|y_{1}-y_{2}|^{n-\alpha'-\beta '}=J_{10}|y_{1}-y_{2}|^{n-(n+\alpha)q}, \quad i=1,2,\ldots,n-1. $$

So by (3.4), we know

$$ I_{1}\leq J_{9}L_{p}[f,\Omega]|y_{1}-y_{2}| {\sum_{i=1}^{n-1}{J_{10}}^{\frac{1}{q}}|y_{1}-y_{2}|^{\frac {n-(n+\alpha)q}{q}}} \leq J_{11}L_{p}[f,\Omega] {|y_{1}-y_{2}|^{\beta}}, $$
(3.5)

where \(\beta=1-\alpha-n/p\), and from \(p>n/(1-\alpha)\), we have \(0<\beta=1-\alpha-n/p=1-\alpha-n/p<1\).

In addition

$$\begin{aligned} I_{2} \leq&J_{12} \sum_{j=1}^{k-1} \sum_{i=1}^{j} \biggl[ \int_{\Omega}\bigl\vert D^{j}f(x)\bigr\vert ^{p}|dx| \biggr]^{\frac{1}{p}} \biggl[ \int _{\Omega}\frac{|dx|}{|x-y_{1}|^{(n+\alpha-i)q}} \biggr]^{\frac {1}{q}}|y_{1}-y_{2}| \\ =&J_{12}|y_{1}-y_{2}| \Biggl\{ \biggl[ \int_{\Omega }\bigl\vert Df(x)\bigr\vert ^{p}|dx| \biggr]^{\frac{1}{p}} \biggl[ \int_{\Omega}\frac {|dx|}{|x-y_{1}|^{(n+\alpha-1)q}} \biggr]^{\frac{1}{q}} \\ &{}+ \sum_{i=1}^{2} \biggl[ \int_{\Omega }\bigl|D^{2}f(x)\bigr|^{p}|dx| \biggr]^{\frac{1}{p}} \biggl[ \int_{\Omega}\frac {|dx|}{|x-y_{1}|^{(n+\alpha-i)q}} \biggr]^{\frac{1}{q}} \\ &{}+\cdots \\ &{}+\sum_{i=1}^{k-1} \biggl[ \int_{\Omega }\bigl|D^{k-1}f(x)\bigr|^{p}|dx| \biggr]^{\frac{1}{p}} \biggl[ \int_{\Omega }\frac{|dx|}{|x-y_{1}|^{(n+\alpha-i)q}} \biggr]^{\frac{1}{q}} \Biggr\} . \end{aligned}$$

Again from \(1< q< n/(n+\alpha-1)\). Thus, for all \(1\leq i\leq k-1\) (\(k< n\)), we have

$${(n+\alpha-i)q< \frac{(n+\alpha-i)n}{(n+\alpha-1)}\leq n}. $$

Thus, for all \(i=1,2,\ldots,k-1\) (\(k< n\)), the integral \(\int_{\Omega }|x-y_{1}|^{-(n+\alpha-i)q}|dx|\) is convergent. So we obtain

$$ I_{2}\leq J_{13}L_{p}[f,\Omega]|y_{1}-y_{2}| \leq J_{14}L_{p}[f,\Omega ]|y_{1}-y_{2}|^{\beta}. $$
(3.6)

And again from the Hölder inequality, we have

$$\begin{aligned} I_{3} \leq&J_{15}L_{p}[f,\Omega]|y_{1}-y_{2}|^{\alpha} {\sum_{j=0}^{k-1} \biggl[ \int_{\Omega}\frac{|dx|}{|x-y_{1}|^{\alpha q}|x-y_{2}|^{(n+\alpha-j-1)q}} \biggr]^{\frac{1}{q}}} \\ =&J_{15}L_{p}[f,\Omega]|y_{1}-y_{2}|^{\alpha} {\sum_{j=0}^{k-1}{I_{3}}_{j}^{\frac{1}{q}}}, \end{aligned}$$

where \({I_{3}}_{j}=\int_{\Omega}|x-y_{1}|^{-\alpha q}|x-y_{2}|^{-(n+\alpha-j-1)q}|dx|\).

When \(j=0\), \({I_{3}}_{0}=\int_{\Omega}|x-y_{1}|^{-\alpha q}|x-y_{2}|^{-(n+\alpha-1)q}|dx|\). Let \(\alpha'=\alpha q\), \(\beta '=(n+\alpha-1)q\). From \(1< q< n/(n+\alpha-1)\), we get

$$\begin{aligned}& 0< \alpha'=\alpha q< \frac{\alpha n}{n+\alpha-1}< n, \\& 0< \beta'=(n+\alpha-1)q< n. \end{aligned}$$

Again from \(1/2\leq\alpha<1\), we have

$$\alpha'+\beta'=(n+2\alpha-1)q>nq>n. $$

Thus by Lemma 2.4, we obtain

$${{I_{3}}_{0}\leq J_{16}|y_{1}-y_{2}|^{n-\alpha'-\beta '}=J_{16}|y_{1}-y_{2}|^{n-(n+2\alpha-1)q}}. $$

When \(j\neq0\), that is, \(j=1,2,\ldots,k-1\), \(k< n\),

$$\alpha q+(n+\alpha-j-1)q< \frac{(n+\alpha-1+\alpha-j)n}{n+\alpha -1}=\biggl(1+\frac{\alpha-j}{n+\alpha-1} \biggr)n< n. $$

So \({I_{3}}_{j}\) (\(j=1,2,\ldots,k-1\), \(k< n\)) is convergent, thus there is no harm to suppose \({I_{3}}_{j}\leq J_{17}\).

Therefore from the above, we obtain

$$\begin{aligned} I_{3} \leq&J_{15}L_{p}[f,\Omega]|y_{1}-y_{2}|^{\alpha} {\Biggl[{I_{3}}_{0}^{\frac{1}{q}}+\sum _{j=1}^{k-1}{I_{3}}_{j}^{\frac {1}{q}} \Biggr]} \\ =& {J_{15}L_{p}[f,\Omega]|y_{1}-y_{2}|^{\alpha } \bigl[J_{16}^{\frac{1}{q}}|y_{1}-y_{2}|^{\frac{n-(n+2\alpha -1)q}{q}}+(k-1)J_{17}^{\frac{1}{q}} \bigr]} \\ \leq& {J_{18}L_{p}[f,\Omega]|y_{1}-y_{2}|^{\alpha +\frac{n-(n+2\alpha-1)q}{q}}+J_{19}L_{p}[f, \Omega ]|y_{1}-y_{2}|^{\alpha}} \\ =& {J_{18}L_{p}[f,\Omega]|y_{1}-y_{2}|^{1-\alpha-\frac {n}{p}}+J_{19}L_{p}[f, \Omega]|y_{1}-y_{2}|^{2\alpha-1-\frac {n}{p}}|y_{1}-y_{2}|^{1-\alpha-\frac{n}{p}}}. \end{aligned}$$

Again when \(1/2\leq\alpha<1\), we know \(\alpha-(1-\alpha )-n/p=2\alpha-1+n/p>0\) and Ω is bounded. Thus we know that \(|y_{1}-y_{2}|^{2\alpha-1-n/p}\) is bounded. So we have

$$ I_{3}\leq {J_{20}L_{p}[f,\Omega]|y_{1}-y_{2}|^{1-\alpha -n/p}}= {J_{20}L_{p}[f,\Omega]|y_{1}-y_{2}|^{\beta}}, $$
(3.7)

where \(\beta=1-\alpha-n/p\).

Thus from (3.3), (3.5), (3.6), and (3.7), we can get

$$\bigl|T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2})\bigr|\leq {(J_{11}+J_{14}+J_{20})L_{p}[f, \Omega]|y_{1}-y_{2}|^{\beta }}=J_{21}L_{p}[f, \Omega]|y_{1}-y_{2}|^{\beta}. $$

In addition, from the above proof process, we know that \(J_{21}\) is only dependent on n, p, α, Ω. Taking \(M_{5}=J_{21}\), then, when \(1/2\leq\alpha<1\), for all \(y_{1},y_{2}\in\Omega\), we can obtain

$$\bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \leq M_{5}(n,p,\alpha ,\Omega)L_{p}[f, \Omega]|y_{1}-y_{2}|^{\beta}. $$

(2) When \(0<\alpha<1/2\), because of \(p_{1}< p\), \(f\in L^{p}(\Omega )\), we know \(f\in L^{p_{1}}(\Omega)\). Let \(q_{1}\) satisfies \(1/p_{1}+1/q_{1}=1\). From \(n/(1-\alpha)< p_{1}< n/(1-2\alpha)\), thus we get \(n/(n+2\alpha-1)< q_{1}< n/(n+\alpha-1)\). So similar to the text in front of the discussion of \(I_{1}\), obviously, we can get \(0<\alpha',\beta'<n\) and \(\alpha'+\beta'=(n+\alpha)q_{1}>n\).

Thus from Lemma 2.4, we have

$$ I_{1}\leq {J_{11}'L_{p_{1}}[f, \Omega]|y_{1}- y_{2}|^{\gamma}}, $$
(3.8)

where \(\gamma=1-\alpha-n/p_{1}\), \(0<\gamma<1\).

Again completely analogous to the discussion in (1), we have

$$ I_{2}\leq {J_{14}'L_{p_{1}}[f, \Omega ]|y_{1}-y_{2}|^{\gamma}}. $$
(3.9)

In addition, similar to the text in front of the discussion of \(I_{3}\), we have

$$I_{3}\leq {J_{15}'L_{p_{1}}[f, \Omega ]|y_{1}-y_{2}|^{\alpha}\sum _{j=0}^{k-1}{I_{3}}_{j}^{\frac{1}{q_{1}}}}, $$

where \({I_{3}}_{j}=\int_{\Omega}|x-y_{1}|^{-\alpha q_{1}}|x-y_{2}|^{-(n+\alpha-j-1)q_{1}}|dx|\).

When \(j=0\), \(0<\alpha'=\alpha q_{1}<n\), \(0<\beta '=(n+\alpha-1)q_{1}<n\). Thus

$$\alpha'+\beta'=(n+2\alpha-1)q_{1}>(n+2 \alpha-1)\frac{n}{n+2\alpha-1}=n. $$

Thus from Lemma 2.4, we can get

$${{I_{3}}_{0}\leq J_{16}'|y_{1}-y_{2}|^{n-(n+2\alpha-1)q_{1}}}. $$

When \(j\neq0\), again completely analogous to the discussion in (1), we know \({I_{3}}_{j}\) is convergent. So there is no harm to suppose \({I_{3}}_{j}\leq J_{17}'\).

Thus, similarly, we have

$$ I_{3}\leq {J_{20}'L_{p_{1}}[f, \Omega ]|y_{1}-y_{2}|^{\gamma}}. $$
(3.10)

Therefore from (3.3), (3.8), (3.9), and (3.10), we obtain

$$\bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \leq {J_{21}'L_{p_{1}}[f, \Omega]|y_{1}-y_{2}|^{\gamma}}. $$

Taking \(M_{6}=J_{21}'\), we can get

$$\bigl\vert T_{\Omega}[f](y_{1})-T_{\Omega}[f](y_{2}) \bigr\vert \leq M_{6}(n,p,\alpha ,\Omega)L_{p_{1}}[f, \Omega]|y_{1}-y_{2}|^{\gamma}. $$

 □

4 The stability and error estimate of generalized Teodorescu operator

Theorem 4.1

Let Ω be as stated above, \(f(x)\in{C^{(r)}(\Omega,{\mathcal {A}}_{n}(R))}\), \(D^{j}{f(x)}\in{ L^{p}(\Omega)}\) (\(j=0,1,\ldots, k-1\)), \(p>n/(1-\alpha)\), \(y\in\Omega\), \(\rho_{0}\) is a given constant, and \(\omega\in B(\rho_{0})\). Then we have

$$\bigl\vert \bigl(T_{\Omega_{\omega}}[f]\bigr) (y)-\bigl(T_{\Omega}[f] \bigr) (y)\bigr\vert \leq M_{7}(n,p,\alpha,\Omega)L_{p}[f, \Omega]\|\omega\|_{\partial\Omega }^{\gamma}, $$

where \(\gamma=(1-\alpha)/n-1/p>0\).

Proof

First

$$\begin{aligned}& \bigl\vert \bigl(T_{\Omega_{\omega}}[f]\bigr) (y)-\bigl(T_{\Omega}[f] \bigr) (y)\bigr\vert \\& \quad = \Biggl\vert \sum_{j=0}^{k-1} \int_{E_{1}}\frac {A_{j+1}}{\omega_{n}}\frac{D^{j}f(x)(\bar{x}-\bar {y})^{j+1}}{|x-y|^{n+\alpha}}\, dx- \sum _{j=0}^{k-1} \int_{E_{2}}\frac{A_{j+1}}{\omega_{n}}\frac {D^{j}f(x)(\bar{x}-\bar{y})^{j+1}}{|x-y|^{n+\alpha}}\, dx\Biggr\vert \\& \quad \leq \Biggl\vert \sum_{j=0}^{k-1} \int_{E_{1}}\frac {A_{j+1}}{\omega_{n}}\frac{D^{j}f(x)(\bar{x}-\bar {y})^{j+1}}{|x-y|^{n+\alpha}}\, dx\Biggr\vert + \Biggl\vert \sum_{j=0}^{k-1} \int_{E_{2}}\frac{A_{j+1}}{\omega_{n}}\frac {D^{j}f(x)(\bar{x}-\bar{y})^{j+1}}{|x-y|^{n+\alpha}}\, dx\Biggr\vert \\& \quad =I_{1}+I_{2}, \end{aligned}$$
(4.1)

where \(E_{1}\), \(E_{2}\) are the domains as in Lemma 2.5.

Then from Ω, \(E_{1}\) being a bounded domain, \(|x-y|^{jq}\) is bounded. Again by the Hölder inequality, we have

$$\begin{aligned} I_{1} \leq& {J_{22}\sum_{j=0}^{k-1} \biggl[ \int _{E_{1}}\bigl\vert D^{j}f(x)\bigr\vert ^{p}|dx| \biggr]^{\frac{1}{p}} \biggl[ \int _{E_{1}}\frac{|x-y|^{(j+1)q}}{|x-y|^{(n+\alpha)q}}|dx| \biggr]^{\frac {1}{q}}} \\ \leq& {J_{23}L_{p}[f,\Omega]\sum _{j=0}^{k-1} \biggl[ \int_{E_{1}}\frac{|x-y|^{jq}}{|x-y|^{(n+\alpha-1)q}}|dx| \biggr]^{\frac{1}{q}}} \\ \leq& {J_{24}L_{p}[f,\Omega]\sum _{j=0}^{k-1} \biggl[ \int_{E_{1}}\frac{1}{|x-y|^{(n+\alpha-1)q}}|dx| \biggr]^{\frac {1}{q}}} \\ \leq& {J_{25}L_{p}[f,\Omega] \biggl[ \int_{E_{1}}\frac {1}{|x-y|^{(n+\alpha-1)q}}|dx| \biggr]^{\frac{1}{q}}}. \end{aligned}$$
(4.2)

Taking \({E_{1}}_{1}=E_{1}\cap\{x\mid|x-y|\geq\|\omega\| _{\partial\Omega}^{\frac{1}{n}}\}\), \({E_{1}}_{2}=E_{1}\cap\{x\mid|x-y|<\| \omega\|_{\partial\Omega}^{\frac{1}{n}}\}\), then obviously \(E_{1}={E_{1}}_{1}\cup{E_{1}}_{2}\), \({E_{1}}_{1}\cap {E_{1}}_{2}=\emptyset\). Thus

$$\begin{aligned}& \biggl[ \int_{E_{1}}\frac{1}{|x-y|^{(n+\alpha -1)q}}|dx| \biggr]^{\frac{1}{q}} \\& \quad = \biggl[ \int_{{E_{1}}_{1}}\frac{1}{|x-y|^{(n+\alpha -1)q}}|dx|+ \int_{{E_{1}}_{2}}\frac{1}{|x-y|^{(n+\alpha -1)q}}|dx| \biggr]^{\frac{1}{q}} \\& \quad \leq \biggl[ \int_{{E_{1}}_{1}}\frac{1}{\|\omega\| _{\partial\Omega}^{\frac{1}{n}(n+\alpha-1)q}}|dx|+ \int _{{E_{1}}_{2}}\frac{1}{|x-y|^{(n+\alpha-1)q}}|dx| \biggr]^{\frac{1}{q}}. \end{aligned}$$
(4.3)

In addition, from Lemma 2.5, we have

$$\begin{aligned}& { \int_{{E_{1}}_{1}}\frac{1}{\|\omega\|_{\partial \Omega}^{\frac{1}{n}(n+\alpha-1)q}}|dx|} \\& \quad = {\|\omega\|_{\partial\Omega}^{\frac{1}{n}(n+\alpha -1)q}A_{{E_{1}}_{1}}}\leq {\| \omega\|_{\partial\Omega }^{-\frac{1}{n}(n+\alpha-1)q}A_{{E_{1}}_{1}\cup{E_{1}}_{2}}} \\& \quad \leq {C\|\omega\|_{\partial\Omega}^{[1-\frac {1}{n}(n+\alpha-1)q]}}= {C\|\omega \|_{\partial\Omega }^{\frac{1}{n}[n-nq+(1-\alpha)q]}}. \end{aligned}$$
(4.4)

Again when \(p>n/(1-\alpha)\), \(1< q< n/(n+\alpha-1)\). Thus \((n+\alpha -1)q-(n-1)<1\). So from the local generalized spherical coordinate, we obtain

$$\begin{aligned}& { \int_{{E_{1}}_{2}}\frac{1}{|x-y|^{(n+\alpha -1)q}}|dx|} \\& \quad \leq {J_{26} \int_{0}^{\|\omega\|_{\partial\Omega }^{\frac{1}{n}}}\frac{1}{\rho_{0}^{[(n+\alpha-1)q-(n-1)]}}\, d\rho _{0}}= {J_{26}\|\omega\|_{\partial\Omega}^{\frac {1}{n}[n-nq+(1-\alpha)q]}}. \end{aligned}$$
(4.5)

Thus from (4.2)-(4.5) and \(1/p+1/q=1\), we get

$$\begin{aligned} I_{1} \leq& J_{25}L_{p}[f,\Omega] \bigl[C\|\omega \| _{\partial\Omega}^{\frac{1}{n}[n-nq+(1-\alpha)q]}+ J_{26}\|\omega\|_{\partial\Omega}^{\frac{1}{n}[n-nq+(1-\alpha)q]} \bigr]^{\frac{1}{q}} \\ \leq& {J_{27}L_{p}[f,\Omega]\|\omega\|_{\partial \Omega}^{\frac{n-nq+(1-\alpha)q}{nq}}} \\ =& {J_{27}L_{p}[f,\Omega]\|\omega\|_{\partial\Omega }^{\frac{(1-\alpha)p-n}{np}}} \\ =& {J_{27}L_{p}[f,\Omega]\|\omega\|_{\partial\Omega }^{\gamma}}, \end{aligned}$$
(4.6)

where \(\gamma=(1-\alpha)/n-1/p>0\).

Similarly, we have

$$ I_{2}\leq {J_{28}L_{p}[f,\Omega]\|\omega \|_{\partial \Omega}^{\gamma}}. $$
(4.7)

Therefore from (4.1), (4.6), and (4.7), we obtain

$$\bigl\vert \bigl(T_{\Omega_{\omega}}[f]\bigr) (y)-\bigl(T_{\Omega}[f] \bigr) (y)\bigr\vert \leq J_{29}L_{p}[f,\Omega]\|\omega \|_{\partial\Omega}^{\gamma}. $$

We take \(M_{7}=J_{29}\), that is,

$$\bigl\vert \bigl(T_{\Omega_{\omega}}[f]\bigr) (y)-\bigl(T_{\Omega}[f] \bigr) (y)\bigr\vert \leq M_{7}L_{p}[f,\Omega]\|\omega \|_{\partial\Omega}^{\gamma}. $$

From the above proof process, we know that \(M_{7}=J_{29}\) is only dependent on n, p, α, Ω. □