By means of the idea as stated above, we suppose
$$\begin{aligned}& \xi=z_{1}+z_{2}j\in\mathbb{H},\quad z_{1}=x_{0}+ix_{1},z_{2}=x_{2}+ix_{3} \in\mathbb{C}, \\& \eta=\varsigma_{1}+\varsigma_{2}j\in\mathbb{H}, \quad \varsigma _{1}=y_{0}+iy_{1},\varsigma_{2}=y_{2}+iy_{3} \in\mathbb{C}, \\& f(\xi)=f(z_{1},z_{2})=u_{1}(z_{1},z_{2})+u_{2}(z_{1},z_{2})j, \quad u_{1}=f_{0}+if_{1},u_{2}=f_{2}+if_{3} \in\mathbb{C}, \\& g(\xi)=g(z_{1},z_{2})=v_{1}(z_{1},z_{2})+v_{2}(z_{1},z_{2})j, \quad v_{1}=g_{0}+ig_{1},v_{2}=g_{2}+ig_{3} \in\mathbb{C}. \end{aligned}$$
Then system (1.1) can be written as
$$ \mbox{(1.1)}\quad \Longleftrightarrow\quad \left \{ \begin{array}{l} \partial_{\bar{z}_{1}}u_{1}+\partial_{z_{2}}\bar{u}_{2}=0, \\ \partial_{\bar{z}_{2}}u_{1}-\partial_{z_{1}}\bar{u}_{2}=0 \end{array} \right .\quad \Longleftrightarrow\quad {{}^{\psi} D[f]}=0. $$
(3.1)
Moreover, if the pair \((u_{1},u_{2})\) of continuously differentiable (up to the second order) complex-valued functions give a solution of system (1.1) then
$$\Delta_{\mathbb{R}^{4}}u_{1} \cong\Delta_{\mathbb{C}^{2}}u_{1}=4 \bigl(\partial_{z_{1}\bar {z}_{1}}^{2}+\partial_{z_{2}\bar{z}_{2}}^{2} \bigr)u_{1}=0. $$
A similar observation is valid for \(\bar{u}_{2}\), so \(u_{1}\), \(\bar {u}_{2}\) are complex-valued harmonic functions, i.e. the set of solutions of system (1.1) contains all holomorphic functions of two complex variables.
Similarly, system (1.2) can be written as
$$ \mbox{(1.2)}\quad \Longleftrightarrow\quad \left \{ \begin{array}{l} 2\partial_{\bar{z}_{1}}u_{1}+2\partial_{z_{2}}\bar{u}_{2}=v_{1}, \\ 2\partial_{\bar{z}_{2}}u_{1}-2\partial_{z_{1}}\bar{u}_{2}=v_{2} \end{array} \right .\quad \Longleftrightarrow\quad {{}^{\psi} D[f]=g}. $$
(3.2)
The generalized Teodorescu type integral operator \({}^{\psi} T_{\mathbb {C}^{2}}[g]\) can be written as
$$\begin{aligned} {}^{\psi} T_{\mathbb{C}^{2}}[g](z_{1},z_{2}) =&{}^{\psi} T_{\mathbb {C}^{2}}[g]( \xi) \\ =& {-\int_{\mathbb{C}^{2}}\mathcal{K}_{\psi}(\eta-\xi)g(\eta )\, d_{{\mathbb{C}^{2}}_{\eta}}} \\ =& {\frac{1}{2\pi^{2}}\int_{\mathbb{C}^{2}}\frac{(\bar{z}_{1}-\bar {\varsigma}_{1})+(\bar{z}_{2}-\bar{\varsigma}_{2})j}{ (|z_{1}-\varsigma_{1}|^{2}+|z_{2}-\varsigma_{2}|^{2})^{2}}g( \varsigma _{1},\varsigma_{2})\, d_{{\mathbb{C}^{2}}_{\varsigma_{1},\varsigma_{2}}}} \\ =&{}^{\psi} T_{\mathbb{C}^{2}}^{(1)}[g](z_{1},z_{2})+{{}^{\psi} T_{\mathbb {C}^{2}}^{(2)}}[g](z_{1},z_{2})j, \end{aligned}$$
where the Cimmino singular integral operators \({}^{\psi} T_{\mathbb {C}^{2}}^{(1)}[g]\), \({{}^{\psi} T_{\mathbb{C}^{2}}^{(2)}[g]}\) are as follows:
$$\begin{aligned}& {}^{\psi} T_{\mathbb{C}^{2}}^{(1)}[g](z_{1},z_{2}) = {\frac{1}{2\pi ^{2}}\int_{\mathbb{C}^{2}}\frac{(\bar{z}_{1}-\bar{\varsigma}_{1})}{ (|z_{1}-\varsigma_{1}|^{2}+|z_{2}-\varsigma_{2}|^{2})^{2}}g(\varsigma _{1},\varsigma_{2})\, d_{{\mathbb{C}^{2}}_{\varsigma_{1},\varsigma _{2}}}}, \\& {}^{\psi} T_{\mathbb{C}^{2}}^{(2)}[g](z_{1},z_{2}) = {\frac{1}{2\pi ^{2}}\int_{\mathbb{C}^{2}}\frac{(\bar{z}_{2}-\bar{\varsigma}_{2})}{ (|z_{1}-\varsigma_{1}|^{2}+|z_{2}-\varsigma_{2}|^{2})^{2}}g(\varsigma _{1},\varsigma_{2})\, d_{{\mathbb{C}^{2}}_{\varsigma_{1},\varsigma_{2}}}}. \end{aligned}$$
Theorem 3.1
Let
E
be as stated above. If
\(g\in L_{p}(\mathbb {C}^{2},\mathbb{H})\), \(4< p<+\infty\), then we have
-
(1)
\(|{}^{\psi} T_{\mathbb{C}^{2}}[g](\xi)|\leq M_{1}(p)\|g\|_{L_{p}}\), \(\xi \in\mathbb{C}^{2}\cong\mathbb{R}^{4}\),
-
(2)
\({}^{\psi} T_{\mathbb{C}^{2}}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb {H})\cong C_{\beta}(\mathbb{R}^{4},\mathbb{H})\) (\(0<\beta=1-4/p<1\)),
-
(3)
\({}^{\psi} D({}^{\psi} T_{\mathbb{C}^{2}}[g])(\xi)=g(\xi)\), \(\xi\in\mathbb {C}^{2}\cong\mathbb{R}^{4}\).
Proof
(1) First, we have
$$\begin{aligned} \bigl\vert {}^{\psi} T_{\mathbb{C}^{2}}[g](\xi)\bigr\vert \leq& { \frac{1}{2\pi^{2}}\biggl\vert \int_{E}\frac{(\bar{z}_{1}-\bar{\varsigma}_{1})+(\bar{z}_{2}-\bar{\varsigma}_{2})j}{ (|z_{1}-\varsigma_{1}|^{2}+|z_{2}-\varsigma_{2}|^{2})^{2}}g( \eta )\, dE_{\eta}\biggr\vert } \\ &{}+ {\frac{1}{2\pi^{2}}\biggl\vert \int_{\mathbb{C}^{2}-E} \frac{(\bar {z}_{1}-\bar{\varsigma}_{1})+(\bar{z}_{2}-\bar{\varsigma}_{2})j}{ (|z_{1}-\varsigma_{1}|^{2}+|z_{2}-\varsigma_{2}|^{2})^{2}}g(\eta )\, d_{{(\mathbb{C}^{2}-E)}_{\eta}}\biggr\vert } \\ =&O_{1}+O_{2}. \end{aligned}$$
By the Hölder inequality, we have
$$\begin{aligned} O_{1} =& {\frac{1}{2\pi^{2}}\biggl\vert \int_{E} \frac{(\bar{z}_{1}-\bar {\varsigma}_{1})+(\bar{z}_{2}-\bar{\varsigma}_{2})j}{ (|z_{1}-\varsigma_{1}|^{2}+|z_{2}-\varsigma_{2}|^{2})^{2}}g(\eta )\, dE_{\eta}\biggr\vert } \\ \leq& {\frac{1}{2\pi^{2}}\int_{E}\frac{|z_{1}-\varsigma _{1}|+|z_{2}-\varsigma_{2}|}{|\xi-\eta|^{4}}\bigl\vert g(\eta)\bigr\vert \, dE_{\eta}} \\ \leq& {\frac{1}{\pi^{2}}\int_{E}\frac{1}{|\xi-\eta|^{3}}\bigl\vert g(\eta )\bigr\vert \, dE_{\eta}}\leq{\frac{1}{\pi^{2}}\|g \|_{L_{p}} \biggl(\int_{E}\frac {1}{|\xi-\eta|^{3q}}\, dE_{\eta}\biggr)^{\frac{1}{q}}}, \end{aligned}$$
where \(1/p+1/q=1\).
When \(\xi\in\bar{E}\), because of \(4< p<+\infty\), \(1/p+1/q=1\), we have \(1< q<4/3\). Thus \(\int_{E}\frac{1}{|\xi-\eta|^{3q}}\, dE_{\eta}\) is bounded. Hence we have
$${ \biggl(\int_{E}\frac{1}{|\xi-\eta|^{3q}}\, dE_{\eta}\biggr)^{\frac {1}{q}}\leq J_{1}}. $$
When \(\xi\in\mathbb{C}^{2}-\bar{E}\), by Lemma 2.1 and the generalized spherical coordinate, we have
$$ { \biggl(\int_{E}\frac{1}{|\xi-\eta|^{3q}}\, dE_{\eta}\biggr)^{\frac {1}{q}}}\leq{J_{2} \biggl(\int_{d_{0}}^{d_{0}+2} \rho^{3-3q}\, d\rho \biggr)^{\frac{1}{q}}}\leq J_{3}, $$
where \(\rho=|\xi-\eta|\), \(d_{0}=d(\xi, \bar{E})\).
Therefore, for \(\forall\xi\in\mathbb{C}^{2}\cong\mathbb{R}^{4}\), we can obtain
$$O_{1}\leq M'_{1}(p)\|g\|_{L_{p}},\quad \xi\in\mathbb{C}^{2}\cong\mathbb{R}^{4}, $$
where \(M'_{1}(p)=\max\{J_{1}/\pi^{2},J_{3}/\pi^{2}\}\).
For \(\eta\in\mathbb{C}^{2}-E\), we suppose that \(\eta=\frac{\bar {\eta}'}{|\eta'|^{2}}\), then we have \(|\eta'|\leq1\). Thus by \(g\in L_{p}(\mathbb{C}^{2},\mathbb{H})\), similar to the proof as stated above, we have
$$O_{2}\leq M''_{1}(p)\|g \|_{L_{p}}. $$
Therefore, we obtain
$$\bigl\vert {}^{\psi} T_{\mathbb{C}^{2}}[g](\xi)\bigr\vert \leq M_{1}(p)\|g\|_{L_{p}},\quad \xi\in \mathbb{C}^{2} \cong\mathbb{R}^{4}, $$
where \(M_{1}(p)=M'_{1}(p)+M''_{1}(p)\).
(2) For arbitrary \(\xi',\xi''\in\mathbb{C}^{2}\cong\mathbb{R}^{4}\), \(\xi '\neq\xi''\), we have
$$\begin{aligned}& \bigl\vert {}^{\psi} T_{\mathbb{C}^{2}}[g]\bigl(\xi'\bigr)- {{}^{\psi} T_{\mathbb {C}^{2}}}[g]\bigl(\xi''\bigr) \bigr\vert \\& \quad = {\frac{1}{2\pi^{2}}\biggl\vert \int_{\mathbb{C}^{2}} \biggl[ \frac{(\bar {z}_{1}'-\bar{\varsigma}_{1})+(\bar{z}_{2}'-\bar{\varsigma}_{2})j}{|\xi '-\eta|^{4}} -\frac{(\bar{z}_{1}''-\bar{\varsigma}_{1})+(\bar{z}_{2}''-\bar{\varsigma }_{2})j}{|\xi''-\eta|^{4}} \biggr]g(\eta)\, d_{{\mathbb{C}^{2}}_{\eta}} \biggr\vert } \\& \quad \leq {\frac{1}{2\pi^{2}}\biggl\vert \int_{E} \biggl[\frac{(\bar{z}_{1}'-\bar {\varsigma}_{1})+(\bar{z}_{2}'-\bar{\varsigma}_{2})j}{|\xi'-\eta|^{4}} -\frac{(\bar{z}_{1}''-\bar{\varsigma}_{1})+(\bar{z}_{2}''-\bar{\varsigma }_{2})j}{|\xi''-\eta|^{4}} \biggr]g(\eta)\, dE_{\eta}\biggr\vert } \\& \qquad {}+ {\frac{1}{2\pi^{2}}\biggl\vert \int_{\mathbb{C}^{2}-E} \biggl[\frac{(\bar {z}_{1}'-\bar{\varsigma}_{1})+(\bar{z}_{2}'-\bar{\varsigma}_{2})j}{|\xi '-\eta|^{4}} -\frac{(\bar{z}_{1}''-\bar{\varsigma}_{1})+(\bar{z}_{2}''-\bar{\varsigma }_{2})j}{|\xi''-\eta|^{4}} \biggr]g(\eta)\, d_{{(\mathbb{C}^{2}-E)}_{\eta}} \biggr\vert } \\& \quad = O_{3}+O_{4} \end{aligned}$$
and
$$\begin{aligned} O_{3} \leq& {\frac{1}{2\pi^{2}}\biggl\vert \int _{E} \biggl[\frac{(\bar {z}_{1}'-\bar{\varsigma}_{1})}{|\xi'-\eta|^{4}}-\frac{(\bar {z}_{1}''-\bar{\varsigma}_{1})}{|\xi''-\eta|^{4}} \biggr]g( \eta)\, dE_{\eta}\biggr\vert } \\ &{} + {\frac{1}{2\pi^{2}}\biggl\vert \int_{E} \biggl[ \frac{(\bar{z}_{2}'-\bar {\varsigma}_{2})j}{|\xi'-\eta|^{4}}-\frac{(\bar{z}_{2}''-\bar{\varsigma }_{2})j}{|\xi''-\eta|^{4}} \biggr]g(\eta)\, dE_{\eta}\biggr\vert } \\ =&\bigl\vert {}^{\psi} T_{E}^{(1)}[g] \bigl(z_{1}',z_{2}' \bigr)-{{}^{\psi } T_{E}^{(1)}}[g]\bigl(z_{1}'',z_{2}'' \bigr)\bigr\vert +\bigl\vert {}^{\psi } T_{E}^{(2)}[g] \bigl(z_{1}',z_{2}' \bigr)-{{}^{\psi } T_{E}^{(2)}}[g]\bigl(z_{1}'',z_{2}'' \bigr)\bigr\vert \\ =&I_{1}+I_{2}. \end{aligned}$$
(3.3)
Since
$$\begin{aligned}& {\biggl\vert \frac{\bar{z}_{1}'-\bar{\varsigma}_{1}}{|\xi'-\eta|^{4}}-\frac {\bar{z}_{1}''-\bar{\varsigma}_{1}}{|\xi''-\eta|^{4}}\biggr\vert } \\& \quad = \biggl\vert \frac{(\bar{z}_{1}'-\bar{\varsigma}_{1})|\xi''-\eta |^{2}(|z_{1}''-\varsigma_{1}|^{2}+|z_{2}''-\varsigma_{2}|^{2})}{|\xi '-\eta|^{4}|\xi''-\eta|^{4}} \\& \qquad {} - \frac{(|z_{1}'-\varsigma_{1}|^{2}+|z_{2}'-\varsigma _{2}|^{2})|\xi'-\eta|^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})}{|\xi'-\eta |^{4}|\xi''-\eta|^{4}}\biggr\vert \\& \quad \leq {\frac{\vert (\bar{z}_{1}'-\bar{\varsigma}_{1})|\xi''-\eta |^{2}|z_{1}''-\varsigma_{1}|^{2}-|z_{1}'-\varsigma_{1}|^{2}|\xi'-\eta |^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})\vert }{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \qquad {}+ {\frac{\vert (\bar{z}_{1}'-\bar{\varsigma}_{1})|\xi''-\eta |^{2}|z_{2}''-\varsigma_{2}|^{2}-|z_{2}'-\varsigma_{2}|^{2}|\xi'-\eta |^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})\vert }{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \quad = \mathcal{K}_{1}\bigl(\xi',\xi'', \eta\bigr)+\mathcal{K}_{2}\bigl(\xi',\xi'', \eta\bigr). \end{aligned}$$
(3.4)
Thus
$$ I_{1}\leq{\frac{1}{2\pi^{2}}\int_{E} \mathcal{K}_{1}\bigl(\xi',\xi'', \eta \bigr)\bigl\vert g(\eta)\bigr\vert \, dE_{\eta}}+ { \frac{1}{2\pi^{2}}\int_{E}\mathcal{K}_{2}\bigl( \xi',\xi '',\eta\bigr)\bigl\vert g(\eta) \bigr\vert \, dE_{\eta}}=I_{1_{1}}+I_{1_{2}}. $$
(3.5)
Again, because of
$$\begin{aligned}& \mathcal{K}_{1}\bigl(\xi', \xi'',\eta\bigr) \\& \quad = {\frac{\vert (\bar{z}_{1}'-\bar{\varsigma}_{1})|\xi''-\eta |^{2}|z_{1}''-\varsigma_{1}|^{2}-|z_{1}'-\varsigma_{1}|^{2}|\xi'-\eta |^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})\vert }{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \quad = {\frac{\vert (\bar{z}_{1}'-\bar{\varsigma}_{1})|\xi''-\eta |^{2}(z_{1}''-\varsigma_{1})(\bar{z}_{1}''-\bar{\varsigma}_{1}) -(\bar{z}_{1}'-\bar{\varsigma}_{1})(z_{1}'-\varsigma_{1})|\xi'-\eta |^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})\vert }{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \quad = {\frac{|\bar{z}_{1}'-\bar{\varsigma}_{1}|\vert |\xi''-\eta |^{2}(z_{1}''-\varsigma_{1}) -(z_{1}'-\varsigma_{1})|\xi'-\eta|^{2}\vert |\bar{z}_{1}''-\bar {\varsigma}_{1}|}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}} \\& \quad \leq {\frac{|\bar{z}_{1}'-\bar{\varsigma}_{1}| [|\xi''-\eta |^{2}|z_{1}''-z_{1}'| +\vert |\xi''-\eta|^{2}-|\xi'-\eta|^{2}\vert |z_{1}'-\varsigma _{1}| ]|\bar{z}_{1}''-\bar{\varsigma}_{1}|}{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \quad \leq {\frac{|\bar{z}_{1}'-\bar{\varsigma}_{1}| [|\xi''-\eta |^{2}|z_{1}''-z_{1}'| +|\xi''-\xi'| (|\xi''-\eta|+|\xi'-\eta| )|z_{1}'-\varsigma _{1}| ]|\bar{z}_{1}''-\bar{\varsigma}_{1}|}{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \quad \leq {\frac{|\xi'-\eta| [|\xi''-\eta|^{2}|\xi''-\xi'| +|\xi''-\xi'| (|\xi''-\eta|+|\xi'-\eta| )|\xi'-\eta| ]|\xi ''-\eta|}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}} \\& \quad = {\bigl\vert \xi'-\xi''\bigr\vert \sum_{l=1}^{3}\frac{1}{|\xi'-\eta|^{4-l}|\xi''-\eta |^{l}}}. \end{aligned}$$
(3.6)
Thus by (3.5), (3.6), and the Hölder inequality, we have
$$\begin{aligned} \begin{aligned}[b] I_{1_{1}}&\leq {\frac{1}{2\pi^{2}}\int _{E}\sum_{l=1}^{3} \frac{1}{|\xi '-\eta|^{4-l}|\xi''-\eta|^{l}}\bigl\vert g(\eta)\bigr\vert \, dE_{\eta}\bigl\vert \xi'-\xi''\bigr\vert } \\ &\leq {\frac{1}{2\pi^{2}}\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert \sum_{l=1}^{3} \biggl(\int_{E}\frac{1}{|\xi'-\eta|^{(4-l)q}|\xi''-\eta|^{lq}}\, dE_{\eta}\biggr)^{\frac{1}{q}}} \\ &= {\frac{1}{2\pi^{2}}\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert \sum_{l=1}^{3} \bigl(I_{1_{1}}^{(l)}\bigr)^{\frac{1}{q}}}. \end{aligned} \end{aligned}$$
(3.7)
Suppose \(\alpha_{l}=(4-l)q\), \(\beta_{l}=lq\) (\(l=1,2,3\)). By \(1< q<4/3\), we know
$$0<\alpha_{l},\beta_{l}<4, \qquad \alpha_{l}+ \beta_{l}=4q>4. $$
Thus, by Lemma 2.4, for \(l=1,2,3\), we have
$$\begin{aligned} I_{1_{1}}^{(l)} =& {\int_{E} \frac{1}{|\xi'-\eta|^{(4-l)q}|\xi''-\eta |^{lq}}\, dE_{\eta}} \\ \leq& M_{0}(\alpha_{l},\beta_{l})\bigl\vert \xi'-\xi''\bigr\vert ^{4-\alpha_{l}-\beta _{l}} \\ =&M_{0}(\alpha_{l},\beta_{l})\bigl\vert \xi'-\xi''\bigr\vert ^{4-4q}. \end{aligned}$$
(3.8)
Thus, by inequalities (3.7) and (3.8), we obtain
$$\begin{aligned} I_{1_{1}} \leq& {\frac{1}{2\pi^{2}}\|g\|_{L_{p}} \bigl\vert \xi'-\xi''\bigr\vert \sum _{l=1}^{3}\bigl(M_{0}( \alpha_{l},\beta_{l})\bigl\vert \xi'- \xi''\bigr\vert ^{4-4q}\bigr)^{\frac {1}{q}}} \\ \leq&J_{4} {\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert ^{\beta}}, \end{aligned}$$
(3.9)
where \(0<\beta=1+(4-4q)/q=1-4/p<1\).
Next, we discuss \(I_{1_{2}}\).
For arbitrary \(\xi',\xi''\in\mathbb{C}^{2}\cong\mathbb{R}^{4}\), \(\xi'\neq \xi''\), we suppose \(|\xi'-\xi''|=\delta\) and construct a sphere \(B(\xi ',3\delta)\) with the center at \(\xi'\) and radius 3δ. Next we discuss \(I_{1_{2}}\) in two cases.
(i) If \(B(\xi',3\delta)\cap\bar{E}\neq\emptyset\), then we may suppose \(B(\xi',3\delta)\cap\bar{E}=\Omega_{1}\), \(\bar{E}-\Omega_{1}=\Omega_{2}\). Thus we have
$$\begin{aligned} I_{1_{2}} =& {\frac{1}{2\pi^{2}}\int_{E} \mathcal{K}_{2}\bigl(\xi',\xi'', \eta \bigr)\bigl\vert g(\eta)\bigr\vert \, dE_{\eta}} \\ =& {\frac{1}{2\pi^{2}}\int_{\Omega_{1}}\mathcal{K}_{2} \bigl(\xi',\xi'',\eta \bigr)\bigl\vert g( \eta)\bigr\vert \, d{\Omega_{1}}_{\eta}} \\ &+ {\frac{1}{2\pi^{2}}\int_{\Omega_{2}}\mathcal{K}_{2} \bigl(\xi',\xi'',\eta \bigr)\bigl\vert g( \eta)\bigr\vert \, d{\Omega_{2}}_{\eta}} \\ =&I_{1_{2}}^{(1)}+I_{1_{2}}^{(2)}. \end{aligned}$$
(3.10)
Again, by inequality (3.4), the Hölder inequality, and the use of a local generalized spherical coordinate, we have
$$\begin{aligned} I_{1_{2}}^{(1)} \leq& {\frac{1}{2\pi^{2}}\int _{\Omega_{1}}\frac{|\bar {z}_{1}'-\bar{\varsigma}_{1}||\xi''-\eta|^{2}|z_{2}''-\varsigma _{2}|^{2}}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}\bigl\vert g(\eta)\bigr\vert \, d{ \Omega_{1}}_{\eta }} \\ &{}+ {\frac{1}{2\pi^{2}}\int_{\Omega_{1}}\frac{|z_{2}'-\varsigma _{2}|^{2}|\xi'-\eta|^{2}|\bar{z}_{1}''-\bar{\varsigma}_{1}|}{|\xi'-\eta |^{4}|\xi''-\eta|^{4}}\bigl\vert g(\eta)\bigr\vert \, d{\Omega_{1}}_{\eta}} \\ \leq& {\frac{1}{2\pi^{2}}\int_{\Omega_{1}}\frac{1}{|\xi'-\eta |^{3}}\bigl\vert g(\eta)\bigr\vert \, d{\Omega_{1}}_{\eta}}+ { \frac{1}{2\pi^{2}}\int_{\Omega _{1}}\frac{1}{|\xi''-\eta|^{3}}\bigl\vert g( \eta)\bigr\vert \, d{\Omega_{1}}_{\eta}} \\ \leq&J_{5}\|g\|_{L_{p}} { \biggl[ \biggl(\int _{\Omega_{1}}\frac{1}{|\xi '-\eta|^{3q}}\, d{\Omega_{1}}_{\eta} \biggr)^{\frac{1}{q}}+ \biggl(\int_{\Omega_{1}}\frac{1}{|\xi''-\eta|^{3q}}\, d{\Omega_{1}}_{\eta} \biggr)^{\frac{1}{q}} \biggr]} \\ \leq&J_{6}\|g\|_{L_{p}} { \biggl[ \biggl(\int _{0}^{3\delta}\frac{1}{\rho ^{3q-3}}\, d\rho \biggr)^{\frac{1}{q}}+ \biggl(\int_{0}^{4\delta} \frac{1}{\rho ^{3q-3}}\, d\rho \biggr)^{\frac{1}{q}} \biggr]} \\ \leq&J_{7}\|g\|_{L_{p}}\delta^{\frac{4-3q}{q}}=J_{7} \|g\|_{L_{p}}\bigl\vert \xi '-\xi'' \bigr\vert ^{1-\frac{4}{p}} \\ =&J_{7}\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert ^{\beta}. \end{aligned}$$
(3.11)
In addition,
$$\begin{aligned}& \mathcal{K}_{2}\bigl(\xi',\xi'', \eta\bigr) \\& \quad = {\frac{\vert (\bar{z}_{1}'-\bar{\varsigma}_{1})|\xi''-\eta |^{2}|z_{2}''-\varsigma_{2}|^{2}-|z_{2}'-\varsigma_{2}|^{2}|\xi'-\eta |^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})\vert }{|\xi'-\eta|^{4}|\xi ''-\eta|^{4}}} \\& \quad = {\frac{\vert (\bar{z}_{1}''-\bar{\varsigma}_{1}+\bar{z}_{1}'-\bar {z}_{1}'')|\xi''-\eta|^{2}|z_{2}''-\varsigma_{2}|^{2}-|z_{2}'-\varsigma _{2}|^{2}|\xi'-\eta|^{2}(\bar{z}_{1}''-\bar{\varsigma}_{1})\vert }{|\xi '-\eta|^{4}|\xi''-\eta|^{4}}} \\& \quad = \biggl\vert \frac{(\bar{z}_{1}''-\bar{\varsigma}_{1})|\xi''-\eta |^{2}|z_{2}''-\varsigma_{2}|^{2}+(\bar{z}_{1}'-\bar{z}_{1}'')|\xi''-\eta |^{2}|z_{2}''-\varsigma_{2}|^{2}}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}} \\& \qquad {} - \frac{(\bar{z}_{1}''-\bar{\varsigma}_{1})|z_{2}'-\varsigma _{2}|^{2}|\xi'-\eta|^{2}}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}\biggr\vert \\& \quad \leq {\frac{|z_{1}''-\varsigma_{1}|\vert |\xi''-\eta |^{2}|z_{2}''-\varsigma_{2}|^{2}-|z_{2}'-\varsigma_{2}|^{2}|\xi'-\eta |^{2}\vert }{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}} \\& \qquad {} + {\frac{|z_{1}'-z_{1}''||\xi''-\eta|^{2}|z_{2}''-\varsigma _{2}|^{2}}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}}. \end{aligned}$$
Again, because of
$$\begin{aligned}& \bigl\vert \bigl\vert \xi''-\eta\bigr\vert ^{2}\bigl\vert z_{2}''- \varsigma_{2}\bigr\vert ^{2}-\bigl\vert z_{2}'-\varsigma _{2}\bigr\vert ^{2} \bigl\vert \xi'-\eta\bigr\vert ^{2}\bigr\vert \\& \quad = \bigl\vert \bigl\vert \xi''-\eta\bigr\vert ^{2}\bigl\vert z_{2}''- \varsigma_{2}\bigr\vert ^{2}-\bigl\vert \xi''-\eta \bigr\vert ^{2}\bigl\vert z_{2}'-\varsigma_{2}\bigr\vert ^{2} \\& \qquad {}+\bigl\vert \xi''-\eta\bigr\vert ^{2}\bigl\vert z_{2}'-\varsigma_{2} \bigr\vert ^{2}-\bigl\vert z_{2}'-\varsigma _{2}\bigr\vert ^{2}\bigl\vert \xi'-\eta\bigr\vert ^{2}\bigr\vert \\& \quad \leq \bigl\vert \xi''-\eta\bigr\vert ^{2}\bigl\vert \bigl\vert z_{2}''- \varsigma _{2}\bigr\vert ^{2}-\bigl\vert z_{2}'-\varsigma_{2}\bigr\vert ^{2} \bigr\vert +\bigl\vert z_{2}'-\varsigma _{2} \bigr\vert ^{2}\bigl\vert \bigl\vert \xi''- \eta\bigr\vert ^{2}-\bigl\vert \xi'-\eta\bigr\vert ^{2}\bigr\vert \\& \quad \leq \bigl\vert \xi''-\eta\bigr\vert ^{2}\bigl\vert z_{2}''-z_{2}' \bigr\vert \bigl(\bigl\vert z_{2}''- \varsigma _{2}\bigr\vert +\bigl\vert z_{2}'- \varsigma_{2}\bigr\vert \bigr) +\bigl\vert z_{2}'-\varsigma_{2} \bigr\vert ^{2}\bigl\vert \xi''- \xi'\bigr\vert \bigl(\bigl\vert \xi''- \eta\bigr\vert +\bigl\vert \xi'-\eta \bigr\vert \bigr) \\& \quad \leq {\bigl\vert \xi''-\xi'\bigr\vert \sum_{m=0}^{3}\bigl\vert \xi''-\eta\bigr\vert ^{3-m}\bigl\vert \xi'-\eta\bigr\vert ^{m}}. \end{aligned}$$
Thus, we have
$$\begin{aligned}& \mathcal{K}_{2}\bigl(\xi', \xi'',\eta\bigr) \\& \quad \leq {\frac{|\xi''-\eta||\xi''-\xi'|\sum_{m=0}^{3}|\xi''-\eta |^{3-m}|\xi'-\eta|^{m}+|\xi'-\xi''||\xi''-\eta|^{2}|\xi''-\eta |^{2}}{|\xi'-\eta|^{4}|\xi''-\eta|^{4}}} \\& \quad = {\bigl\vert \xi'-\xi''\bigr\vert \Biggl(\frac{2}{|\xi'-\eta|^{4}}+\sum_{l=1}^{3} \frac {1}{|\xi'-\eta|^{4-l}|\xi''-\eta|^{l}} \Biggr)}. \end{aligned}$$
(3.12)
So by (3.10) and (3.12), we obtain
$$\begin{aligned} I_{1_{2}}^{(2)} =& {\frac{1}{2\pi^{2}}\int _{\Omega_{2}}\mathcal {K}_{2}\bigl(\xi', \xi'',\eta\bigr)\bigl\vert g(\eta)\bigr\vert \, d{ \Omega_{2}}_{\eta}} \\ \leq& {\bigl\vert \xi'-\xi''\bigr\vert \frac{1}{\pi^{2}}\int_{\Omega_{2}}\frac{|g(\eta )|}{|\xi'-\eta|^{4}}\, d{ \Omega_{2}}_{\eta}} \\ &{}+ {\bigl\vert \xi'-\xi''\bigr\vert \frac{1}{2\pi^{2}}\int_{\Omega_{2}}\sum _{l=1}^{3}\frac {\vert g(\eta)\vert }{|\xi'-\eta|^{4-l}|\xi''-\eta|^{l}}\, d{\Omega_{2}}_{\eta}} \\ =&{I_{1_{2}}^{(2)}}'+{I_{1_{2}}^{(2)}}''. \end{aligned}$$
(3.13)
First, similar to the method estimating \(I_{1_{1}}\), we have
$$ {I_{1_{2}}^{(2)}}''\leq J_{8}\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert ^{\beta}. $$
(3.14)
Second, when \(\eta\in\Omega_{2}\), \(|\xi'-\eta|>3\delta\), \(|\xi''-\eta |>2\delta\). Thus we have
$$ \begin{aligned} &2\delta\leq\bigl\vert \xi'-\eta\bigr\vert -\bigl\vert \xi'-\xi''\bigr\vert \leq\bigl\vert \xi''-\eta\bigr\vert \leq \bigl\vert \xi'-\xi''\bigr\vert +\bigl\vert \xi '-\eta\bigr\vert =\delta+\bigl\vert \xi'-\eta\bigr\vert , \\ &\delta\leq\bigl\vert \xi''-\eta\bigr\vert -\bigl\vert \xi''-\xi'\bigr\vert \leq\bigl\vert \xi'-\eta\bigr\vert \leq\bigl\vert \xi'- \xi''\bigr\vert +\bigl\vert \xi ''- \eta\bigr\vert =\delta+\bigl\vert \xi''-\eta\bigr\vert . \end{aligned} $$
(3.15)
So we know
$$ \frac{1}{2}\leq\frac{|\xi'-\eta|}{|\xi''-\eta|}\leq\frac{3}{2}. $$
(3.16)
Thus, by (3.16), the Hölder inequality, and Lemma 2.4, we can obtain
$$\begin{aligned} {I_{1_{2}}^{(2)}}' =& {\bigl\vert \xi'-\xi''\bigr\vert \frac{1}{\pi^{2}} \int_{\Omega_{2}}\frac{\vert g(\eta)\vert }{|\xi '-\eta|^{4}}\, d{\Omega_{2}}_{\eta}} \\ \leq& {\bigl\vert \xi'-\xi''\bigr\vert \frac{1}{\pi^{2}}\int_{\Omega_{2}}\frac{2|g(\eta )|}{|\xi'-\eta|^{3}|\xi''-\eta|}\, d{ \Omega_{2}}_{\eta}} \\ \leq& {J_{9}\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert ^{1-\frac{4}{p}}} \\ =& {J_{9}\|g\|_{L_{p}}\bigl\vert \xi'- \xi''\bigr\vert ^{\beta}}. \end{aligned}$$
(3.17)
So, by (3.13), (3.14), and (3.17), we have
$$ I_{1_{2}}^{(2)}\leq J_{10}\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}. $$
(3.18)
Therefore, by (3.10), (3.11), and (3.18), we can obtain
$$ I_{1_{2}}\leq J_{11}\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}. $$
(3.19)
(ii) If \(B(\xi',3\delta)\cap\bar{E}=\emptyset\), then for arbitrary \(\eta\in E\), we have \(|\xi'-\eta|>3\delta\), \(|\xi''-\eta|>2\delta\). Thus similar to the method estimating \(I_{1_{2}}^{(2)}\), we have
$$ I_{1_{2}}\leq J_{12}\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}. $$
(3.20)
So, by (3.19) and (3.20), we have
$$ I_{1_{2}}\leq J_{13}\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}, \quad \xi',\xi''\in \mathbb{R}^{4} \cong\mathbb{C}^{2}, $$
(3.21)
where \(J_{13}=\max\{J_{11},J_{12}\}\).
Thus, to sum up, by (3.5), (3.9), and (3.21), we obtain
$$ I_{1}\leq J_{14}\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}, \quad \xi',\xi''\in\mathbb {R}^{4}\cong\mathbb{C}^{2}. $$
(3.22)
Similarly, we have
$$ I_{2}\leq J_{15}\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}, \quad \xi',\xi''\in\mathbb {R}^{4}\cong\mathbb{C}^{2}. $$
(3.23)
So, by (3.3), (3.22), and (3.23), we obtain
$$ O_{3}\leq M'_{2}(p)\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta},\quad \xi',\xi''\in \mathbb{R}^{4}\cong\mathbb{C}^{2}, $$
(3.24)
where \(M'_{2}(p)=J_{14}+J_{15}\).
For \(\eta\in\mathbb{C}^{2}-E\), we suppose that \(\eta=\frac{\bar {\eta}'}{|\eta'|^{2}}\), then we have \(|\eta'|\leq1\). Thus by \(g\in L_{p}(\mathbb{C}^{2},\mathbb{H})\), similar to the proof as stated above, we have
$$O_{4}\leq M''_{2}(p)\|g \|_{L_{p}}\bigl\vert \xi'-\xi'' \bigr\vert ^{\beta},\quad \xi',\xi'' \in \mathbb{R}^{4}\cong\mathbb{C}^{2}. $$
Therefore, for arbitrary \(\xi',\xi''\in\mathbb{C}^{2}\cong\mathbb {R}^{4}\), \(\xi'\neq\xi''\), we obtain
$$\bigl\vert {}^{\psi} T_{\mathbb{C}^{2}}[g]\bigl(\xi'\bigr)- {{}^{\psi} T_{\mathbb{C}^{2}}}[g]\bigl(\xi ''\bigr) \bigr\vert \leq M_{2}(p)\|g\|_{L_{p}}\bigl\vert \xi'-\xi''\bigr\vert ^{\beta}, \quad \xi',\xi''\in \mathbb{R}^{4} \cong\mathbb{C}^{2}, $$
where \(M_{2}(p)=M'_{2}(p)+M''_{2}(p)\), i.e.
\({}^{\psi} T_{\mathbb {C}^{2}}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb{H})\cong C_{\beta }(\mathbb{R}^{4},\mathbb{H})\) (\(0<\beta<1\)).
(3) For arbitrary \(\varphi\in C_{0}^{\infty}(\mathbb{C}^{2},\mathbb {H})\), there exists a bounded closed set \(Q\subset\mathbb{C}^{2}\), such that \(\overline{\operatorname{supp}\varphi}\subset\subset Q\). Thus, by \(T_{\mathbb {C}^{2}}[g](\infty)=0\), Definition 2.1, Lemma 2.3, and the Fubini theorem, we have
$$\begin{aligned}& {\int_{\mathbb{C}^{2}}[\varphi]{{}^{\psi} D}(\xi){{}^{\psi} T_{\mathbb {C}^{2}}[g]}( \xi)\, d_{{\mathbb{C}^{2}}_{\xi}}} \\& \quad = {\lim_{d\rightarrow\infty}\int_{Q}[ \varphi]{{}^{\psi} D}(\xi){{}^{\psi } T_{\mathbb{C}^{2}}[g]}(\xi)\, dQ_{\xi}} \\& \quad = - {\lim_{d\rightarrow\infty}\int_{Q}[ \varphi]{{}^{\psi} D}(\xi)\int_{\mathbb{C}^{2}}\mathcal{K}_{\psi}( \eta-\xi)g(\eta)\, d_{{\mathbb {C}^{2}}_{\eta}}\, dQ_{\xi}} \\& \quad = {\lim_{d\rightarrow\infty}\int_{\mathbb{C}^{2}}\int _{Q}[\varphi ]{{}^{\psi} D}(\xi)\mathcal{K}_{\psi}( \xi-\eta)\, dQ_{\xi}g(\eta)\, d_{{\mathbb {C}^{2}}_{\eta}}} \\& \quad = {\lim_{d\rightarrow\infty}\int_{\mathbb{C}^{2}} \biggl[\int _{\partial Q}\varphi(\xi)\, d\sigma_{\xi} \mathcal{K}_{\psi}(\xi-\eta)-\varphi(\eta ) \biggr]g(\eta)\, d_{{\mathbb{C}^{2}}_{\eta}}} \\& \quad = - {\int_{\mathbb{C}^{2}}\varphi(\eta)g(\eta)\, d_{{\mathbb{C}^{2}}_{\eta}}} =- {\int_{\mathbb{C}^{2}}\varphi(\xi)g(\xi)\, d_{{\mathbb{C}^{2}}_{\xi}}}, \end{aligned}$$
where \(d=\sup_{\xi',\xi''\in Q}|\xi'-\xi''|\). Hence, in the sense of generalized derivatives, we have \({}^{\psi} D({}^{\psi} T_{\mathbb {C}^{2}}[g])(\xi)=g(\xi)\). □
Remark 3.1
By the process of proof in Theorem 3.1, it is easy to show that \({}^{\psi} T_{\mathbb{C}^{2}}^{(1)}[g], {}^{\psi} T_{\mathbb {C}^{2}}^{(2)}[g]\in C_{\beta}(\mathbb{C}^{2},\mathbb{C})\cong C_{\beta }(\mathbb{R}^{4},\mathbb{C})\) (\(0<\beta<1\)).