On Quaternionic Analysis and a Certain Generalized Fractal-Fractional ψ-Fueter Operator

1. Generalized Fractal-Fractional Derivative

Definition 1.

The fractal derivative of a function f, defined on an interval I, with respect to a fractal measure $\nu (\eta ,t)$ is given by

$$\begin{array}{c}\hfill {\displaystyle \frac{d_{\nu }f\left(t\right)}{d{t}^{\eta }}}:=\underset{\tau \to t}{lim}{\displaystyle \frac{f\left(t\right)-f\left(\tau \right)}{\nu (\eta ,t)-\nu (\eta ,\tau )}},\eta >0.\end{array}$$

If ${\displaystyle \frac{d_{\nu }f\left(t\right)}{d{t}^{\eta }}}$ exists for all $t\in I$ then f is real fractal differentiable on I with order η.

Some well-known cases. If $\nu (h,t)=t$ for all $t\in I$ then ${\displaystyle \frac{d_{\nu }}{d{t}^{\eta }}}={\displaystyle \frac{d}{dt}}$ is the derivative operator. In addition, if $\nu (h,t)=h\left(t\right)$ for all $t\in I$ where $h^{\prime }\left(t\right)>0$ for all $t\in I$ then ${\displaystyle \frac{d_{\nu }f\left(t\right)}{d{t}^{\eta }}}={\displaystyle \frac{f^{\prime }\left(t\right)}{h^{\prime }\left(t\right)}}$ for all $f\in C^1\left(I\right)$

On the other hand, if $\nu (\eta ,t)=t^{\eta }$ for all $t\in I$ then ${\displaystyle \frac{d_{\nu }f\left(t\right)}{d{t}^{\eta }}}$ reduces to the Hausdorff derivative. Another useful fractal measure is $\nu (\eta ,t)=e^{t^{\alpha }}$ for all $t\in I$ and $\alpha \in (0,1]$.

We consider an well-known extension of the previous fractal derivative.

Definition 2.

Given $\beta \in [0,1]$ we present the β-fractal derivative of a function f, defined on an interval I, with respect to a fractal measure $\nu (\eta ,t)$:

$$\begin{array}{c}\hfill {\displaystyle \frac{d_{\nu }^{\beta }f\left(t\right)}{d{t}^{\eta }}}:=\underset{\tau \to t}{lim}{\displaystyle \frac{{\left(f\left(t\right)\right)}^{\beta }-{\left(f\left(\tau \right)\right)}^{\beta }}{\nu (\eta ,t)-\nu (\eta ,\tau )}},\eta >0.\end{array}$$

In order to make our description of the concept of fractal-fractional derivatives to be used precise, we introduce the notion of fractional proportional derivative, following [20].

Definition 3.

Let ${\chi }_0,{\chi }_1:[0,1]\times I$ be continuous functions such that

$$\underset{\sigma \to 0^{+}}{lim}{\chi }_1(\sigma ,t)=1,\underset{\sigma \to 0^{+}}{lim}{\chi }_0(\sigma ,t)=0,\underset{\sigma \to 1^{-}}{lim}{\chi }_1(\sigma ,t)=0,\underset{\sigma \to 1^{-}}{lim}{\chi }_0(\sigma ,t)=1.$$

The proportional derivative of $f\in C^1\left(I\right)$ of order $\sigma \in [0,1]$ is given by

$$\begin{array}{c}\hfill D^{\sigma }f\left(t\right)={\chi }_1(\sigma ,t)f\left(t\right)+{\chi }_0(\sigma ,t)f^{\prime }\left(t\right),\forall t\in I.\end{array}$$

A combination of Definitions 2 and 3 yields.

Definition 4.

Let $\beta \in [0,1]$, the proportional β-fractal derivative of $f:I\to \mathbb{R}$ with respect to $\nu (\eta ,t)$ and σ is defined to be

$$\begin{array}{c}\hfill {\displaystyle \frac{d_{\nu }^{\sigma ,\beta }f\left(t\right)}{d{t}^{\eta }}}\left(t\right):={\chi }_1(\sigma ,t)f\left(t\right)+{\chi }_0(\sigma ,t){\displaystyle \frac{d_{\nu }^{\beta }f\left(t\right)}{d{t}^{\eta }}},\end{array}$$

if it exists for all $t\in I$.

Remark 1.

Given $\alpha \in (0,1]$ and $k\in \mathbb{N}$ we will consider the k-truncated exponential function defined as follows

$${\displaystyle e{\left(t^{\alpha }\right)}_k:=\sum _{i=0}^k\frac{{\left(t^{\alpha }\right)}^i}{i!}}$$

for all $t\in \mathbb{R}$. For $k=1$ we have $e{\left(t^{\alpha }\right)}_1=1+t^{\alpha }$ and for $k=\infty$ we have $e{\left(t^{\alpha }\right)}_{\infty }=e^{t^{\alpha }}$.

Remark 2.

Important particular case, when the proportional and fractal measure in Definition 4 are given by

$${\chi }_1(\sigma ,t)=1-\sigma ,{\chi }_0(\sigma ,t)=\sigma ,\nu (k,t)=e{\left(t^{\alpha }\right)}_k$$

for all $\sigma \in [0,1]$ and $t\in I$ allows to introduce some cases of generalized fractal-fractional derivative to consider. For $f\in C^1\left(I\right)$ we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{\sigma ,\beta }}{d{t}_{\alpha ,k}}}f\left(t\right):=(1-\sigma )f\left(t\right)+\sigma {\displaystyle \frac{{\left(f^{\beta }\right)}^{\prime }\left(t\right)}{e{\left(t^{\alpha }\right)}_k^{\prime }}},\end{array}$$

for all $t\in I$. The particular cases $k=1,\infty$ reduces to

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{\sigma ,\beta }}{d{t}_{\alpha ,1}}}f\left(t\right)=(1-\sigma )f\left(t\right)+\sigma {\displaystyle \frac{{\left(f^{\beta }\right)}^{\prime }\left(t\right)}{\alpha t^{\alpha -1}}},\\ \hfill {\displaystyle \frac{d^{\sigma ,\beta }}{d{t}_{\alpha ,\infty }}f\left(t\right)=(1-\sigma )f\left(t\right)+\sigma \frac{{\left(f^{\beta }\right)}^{\prime }\left(t\right)}{\alpha t^{\alpha -1}{e}^{t^{\alpha }}},}\end{array}$$

where clearly the conditions $\alpha \in (0,1]$ and $t>0$ are necessary.

Addressing the issue $\sigma =\alpha$ requires that the case $\alpha =0$ should be omitted.

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{\alpha ,\beta }}{d{t}_{\alpha ,1}}}f\left(t\right)=(1-\alpha )f\left(t\right)+{\displaystyle \frac{{\left(f^{\beta }\right)}^{\prime }\left(t\right)}{t^{\alpha -1}}},\\ \hfill {\displaystyle \frac{d^{\alpha ,\beta }}{d{t}_{\alpha ,\infty }}f\left(t\right)=(1-\alpha )f\left(t\right)+\frac{{\left(f^{\beta }\right)}^{\prime }\left(t\right)}{t^{\alpha -1}{e}^{t^{\alpha }}}.}\end{array}$$

The k-truncated exponential function as fractal measure provides the generalized fractal-fractional drivative in much generality

$$\begin{array}{c}\hfill {\displaystyle \frac{d^{\alpha ,\beta }}{d{t}_{\alpha ,k}}}f\left(t\right):=(1-\alpha )f\left(t\right)+{\displaystyle \frac{{\left(f^{\beta }\right)}^{\prime }\left(t\right)}{t^{\alpha -1}e{\left(t^{\alpha }\right)}_{k-1}}}.\end{array}$$

2. Preliminaries on Quaternionic Analysis

We begin by recalling some background and fixing notation that will be used throughout the entire document. For more details, we refer the interested reader to [21,22,23].

A real quaternion is an element of the form $x=x_0+x_1{e}_1+x_2{e}_2+x_3{e}_3,$ where $x_0,x_1,x_2,x_3\in \mathbb{R}$ and the imaginary units $e_1$, $e_2$, $e_3$ satisfy:

$$e_1^2=e_2^2=e_3^2=-1,e_1{e}_2=-e_2{e}_1=e_3,e_2{e}_3=-e_3{e}_2=e_1,e_3{e}_1=-e_1{e}_3=e_2.$$

Quaternions form a skew-field denoted by $\mathbb{H}$. The set $\{1,e_1,e_2,e_3\}$ is the standard basis of $\mathbb{H}$.

The vector part of $x\in \mathbb{H}$ is by definition, $\mathbf{x}:=x_1{e}_1+x_2{e}_2+x_3{e}_3$ while its real part is $x_0:=x_0$. The quaternionic conjugation of x, denoted by $\overline{x}$ is defined by $\overline{x}=:x_0-\mathbf{x}$ and norm the $x\in \mathbb{H}$ is given by

$$\parallel x\parallel :=\sqrt{x_0^2+x_1^2+x_2^3+x_3^2}=\sqrt{x\overline{x}}=\sqrt{\overline{x}x}.$$

The quaternionic scalar product of $x,y\in \mathbb{H}$ is given by

$$\langle x,y\rangle :={\displaystyle \frac{1}{2}}(\overline{x}y+\overline{y}x)={\displaystyle \frac{1}{2}}(x\overline{y}+y\overline{x}).$$

A set of quaternions $\psi =\{{\psi }_0,{\psi }_1,{\psi }_2,{\psi }_2\}$ is called structural set if $\langle {\psi }_k,{\psi }_s\rangle ={\delta }_{k,s}$, for $k,s=0,1,2,3$ and any quaternion x can be rewritten as $x_{\psi }:={\sum }_{k=0}^3{x}_k{\psi }_k$, where $x_k\in \mathbb{R}$ for all k. Notion of structural sets is due to N$\widehat{o}$no [24,25].

Given $q,x\in \mathbb{H}$ we follow the notation used in [21] to write

$${\langle q,x\rangle }_{\psi }=\sum _{k=0}^3{q}_k{x}_k,$$

where $q_k,x_k\in \mathbb{R}$ for all k.

Let $\psi$ an structural set. From now on, we will use the mapping

$$\sum _{k=0}^3{x}_k{\psi }_k\to (x_0,x_1,x_2,x_3).$$

(1)

in essential way.

Given a domain $\Omega \subset \mathbb{H}\cong {\mathbb{R}}^4$ and a function $f:\Omega \to \mathbb{H}$. Then f is written as: $f={\sum }_{k=0}^3{f}_k{\psi }_k$, where $f_k,k=0,1,2,3,$ are $\mathbb{R}$-valued functions. Properties of f are due to properties of all components $f_k$ such as continuity, differentiability, integrability and so on. For example, $C^1(\Omega ,\mathbb{H})$ denotes the set of continuously differentiable $\mathbb{H}$-valued functions defined in $\Omega$.

The left- and the right-$\psi$-Fueter operators are given by $\psi \mathcal{D}\left[f\right]:={\sum }_{k=0}^3{\psi }_k{\partial }_{k}f$ and $\psi {\mathcal{D}}_r\left[f\right]:={\sum }_{k=0}^3{\partial }_{k}f{\psi }_k$, for all $f\in C^1(\Omega ,\mathbb{H})$, respectively, where ${\displaystyle {\partial }_{k}f=\frac{\partial f}{\partial x_k}}$ for all k.

Let $\partial \Omega$ be a $3-$dimensional smooth surface. Then recall the Borel-Pompieu and differential and integral versions of Stokes’ formulas

$$\begin{array}{c}{\int }_{\partial \Omega }(K_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }f\left(\tau \right)+g\left(\tau \right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x))\hfill \\ \hfill & -{\int }_{\Omega }(K_{\psi }(y-x)\psi \mathcal{D}\left[f\right]\left(y\right)+\psi {\mathcal{D}}_r\left[g\right]\left(y\right)K_{\psi }(y-x))dy\hfill \\ \hfill =& \left\{\begin{array}{cc}f\left(x\right)+g\left(x\right),\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega },\hfill \end{array}\right.\hfill \end{array}$$

(2)

for all $f,g\in C^1(\Omega ,\mathbb{H})$.

$$\begin{array}{cc}\hfill d\left(g{\sigma }_x^{\psi }f\right)=& \left(g\psi \mathcal{D}\left[f\right]+\psi {\mathcal{D}}_r\left[g\right]f\right)dx,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\int }_{\partial \Omega }g{\sigma }_x^{\psi }f=& {\int }_{\Omega }\left(g\psi \mathcal{D}\left[f\right]+\psi {\mathcal{D}}_r\left[g\right]f\right)dx,\hfill \end{array}$$

(4)

for all $f,g\in C^1(\overline{\Omega },\mathbb{H})$. Here d represents the exterior differentiation operator, $dx$ is the differential form of the 4-dimensional volume in ${\mathbb{R}}^4$ and

$${\sigma }_x^{\psi }:=-sgn\psi \left(\sum _{k=0}^3{(-1)}^k{\psi }_{k}d{\widehat{x}}_k\right)$$

is the quaternionic differential form of the 3-dimensional volume in ${\mathbb{R}}^4$ according to $\psi$, where $d{\widehat{x}}_k=d{x}_0\wedge d{x}_1\wedge d{x}_2\wedge d{x}_3$ omitting factor $d{x}_k$. In addition, $sgn\psi$ is 1, or $-1$, if $\psi$ and ${\psi }_{std}:=\{1,\mathbf{i},\mathbf{j},\mathbf{k}\}$ have the same orientation, or not, respectively. Note that, $|{\sigma }_x^{\psi }|=d{S}_3$ is the differential form of the 3-dimensional volume in ${\mathbb{R}}^4$ and write ${\sigma }_x={\sigma }_x^{{\psi }_{std}}$. Let us recall that the $\psi$-Cauchy Kernel is given by

$$K_{\psi }(\tau -x)={\displaystyle \frac{1}{2{\pi }^2}}{\displaystyle \frac{\overline{{\tau }_{\psi }-x_{\psi }}}{|{\tau }_{\psi }-x_{\psi }{|}^4}}.$$

3. A Function Theory Generated by a $\beta$-Proportional Fractal Fueter Operator

Let us extend Definition 4 to a quaternionic differential operator associate to an arbitrary structural set $\psi$.

Definition 5.

Let $\Omega \subset \mathbb{H}$ a domain. Fix $\beta =({\beta }_0,{\beta }_1,{\beta }_2,{\beta }_3)\in {[0,1]}^4$ and $\nu =({\nu }_0,{\nu }_1,{\nu }_2,{\nu }_3)$ where ${\nu }_k({\eta }_k,x_k)$ is a fractal measure for $k=0,1,2,3$ according to Definition 1. Denote ${\chi }_1=({\chi }_{0,1},{\chi }_{1,1},{\chi }_{2,1},{\chi }_{3,1})$, ${\chi }_0=({\chi }_{0,0},{\chi }_{1,0},{\chi }_{2,0},{\chi }_{3,0})$ and $\sigma =({\sigma }_0,{\sigma }_1,{\sigma }_2,{\sigma }_3)\in {[0,1]}^4$ where ${\chi }_{k,1}({\sigma }_k,x_k)$ and ${\chi }_{k,0}({\sigma }_k,x_k)$ and are given by Definition 3 on coordinate $x_k$ for $k=0,1,2,3$.

Let $f:\Omega \to \mathbb{H}$ such that ${\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}f\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}$ exists for all $x\in \Omega$ and all $n=0,1,2,3$. Then, the quaternionic ψ-proportional β-fractal derivative of f with respect to ν and σ, is given by

$$\begin{array}{cc}\hfill \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right):=& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{{\partial }_{{\nu }_n}^{{\sigma }_n,{\beta }_n}f\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}\hfill \\ \hfill =& \sum _{n=0}^3{\psi }_n\left({\chi }_{n,1}({\sigma }_n,x_n)f\left(x\right)+{\chi }_{n,0}({\sigma }_n,x_n){\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}f\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}\right)\hfill \\ \hfill =& \sum _{n=0=m}^3{\psi }_n{\psi }_m\left({\chi }_{n,1}({\sigma }_n,x_n)f_m\left(x\right)+{\chi }_{n,0}({\sigma }_n,x_n){\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}{f}_m\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}\right).\hfill \end{array}$$

Proposition 1.

Given $f\in C^1(\Omega ,\mathbb{H})$ as above let us assume that

$$\begin{array}{c}\hfill {\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\left(x\right):={\int }_0^{x_n}{\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}{f}_m\left(x\right)}{\partial {\left(t_n\right)}^{{\eta }_n}}}dt,{\lambda }_{{\nu }_n}^{{\beta }_n}\left(f\right)\left(x\right):=\sum _{m=0}^3{\psi }_m{\int }_0^{x_n}{\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}{f}_m\left(x\right)}{\partial {\left(t_n\right)}^{{\eta }_n}}}dt\end{array}$$

exist for $n,m=0,1,2,3$. Under conditions ${\chi }_{n,0}({\sigma }_n,x_n)\ne 0$ and ${\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\left(x\right)\ne 0$ for all $x=(x_0,x_1,x_2,x_3)\in \Omega$ and all $m,n=0,1,2,3$ we have

$$\begin{array}{cc}\hfill \psi \mathcal{D}\circ {\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)=& \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)+{\mathcal{E}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)+\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\left(x\right),\hfill \end{array}$$

for all $x\in \Omega$, where

$$\begin{array}{cc}\hfill {\mathcal{E}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right):=& \sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\left({\chi }_{k,0}({\sigma }_k,x_k)\right){\lambda }_{{\nu }_k}^{{\beta }_k}\left(f\right)\left(x\right)\right],\hfill \\ \hfill {\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left(f\right)\left(x\right)=& \sum _{k=0}^3\left({\chi }_{k,0}({\sigma }_k,x_k)\right){\lambda }_{{\nu }_k}^{{\beta }_k}\left(f\right)\left(x\right),\hfill \\ \hfill L_{n,m}\left[f\right]\left(x\right):=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\chi }_{n,0}({\sigma }_n,x_n)}{e^{h_{n,m}\left(x\right)}}}\right)e^{h_{n,m}\left(x\right)},\hfill \\ \hfill h_{n,m}\left(x\right)=& {\int }_0^{x_n}{\displaystyle \frac{{\chi }_{n,1}({\sigma }_n,t_n)}{{\chi }_{n,0}({\sigma }_n,t_n)}}{\displaystyle \frac{f_m}{{\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)}}dt,\hfill \end{array}$$

for $n,m\in \{0,1,2,3\}.$

Proof. 

To simplify notation consider ${\lambda }_n={\lambda }_{{\nu }_n}^{{\beta }_n}$ for all $n=0,1,2,3$. From direct computations we have that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,m}\left(x\right)}{\lambda }_n\left(f_m\right)\left(x\right)\right)=& e^{h_{n,m}\left(x\right)}\left[{\displaystyle \frac{{\chi }_{n,1}({\sigma }_n,x_n)}{{\chi }_{n,0}({\sigma }_n,x_n)}}{\displaystyle \frac{f_m}{{\lambda }_n\left(f_m\right)}}{\lambda }_n\left(f_m\right)\left(x\right)+{\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}{f}_m\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}\right]\hfill \\ \hfill =& {\displaystyle \frac{e^{h_{n,m}\left(x\right)}}{{\chi }_{n,0}({\sigma }_n,x_n)}}\left[{\chi }_{n,1}({\sigma }_n,t_n)f_m+{\chi }_{n,0}({\sigma }_n,x_n){\displaystyle \frac{{\partial }_{{\nu }_n}^{{\beta }_n}{f}_m\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}\right]\hfill \\ \hfill =& {\displaystyle \frac{e^{h_{n,m}\left(x\right)}}{{\chi }_{n,0}({\sigma }_n,x_n)}}{\displaystyle \frac{{\partial }_{{\nu }_n}^{{\sigma }_n,{\beta }_n}{f}_m\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}},\hfill \\ \hfill {\displaystyle \frac{{\partial }_{{\nu }_n}^{{\sigma }_n,{\beta }_n}{f}_m\left(x\right)}{\partial {\left(x_n\right)}^{{\eta }_n}}}=& {\displaystyle \frac{{\chi }_{n,0}({\sigma }_n,x_n)}{e^{h_{n,m}\left(x\right)}}}{\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,m}\left(x\right)}{\lambda }_n\left(f_m\right)\left(x\right)\right).\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill & \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)=\sum _{n=0=m}^3{\psi }_n{\psi }_m{\displaystyle \frac{{\chi }_{n,0}({\sigma }_n,x_n)}{e^{h_{n,m}\left(x\right)}}}{\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,m}\left(x\right)}{\lambda }_n\left(f_m\right)\left(x\right)\right)\hfill \\ \hfill =& \sum _{n=0=m}^3{\psi }_n{\psi }_m{\displaystyle \frac{\partial }{\partial x_n}}\left({\chi }_{n,0}({\sigma }_n,x_n){\lambda }_n\left(f_m\right)\left(x\right)\right))-\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right)\hfill \\ \hfill =& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left({\chi }_{n,0}({\sigma }_n,x_n)\sum _{m=0}^3{\psi }_m{\lambda }_n\left(f_m\right)\left(x\right)\right))-\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right)\hfill \\ \hfill =& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[{\chi }_{n,0}({\sigma }_n,x_n){\lambda }_n\left(f\right)\left(x\right)\right]-\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right)\hfill \\ \hfill =& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\varsigma \left(x\right){\lambda }_n\left(f\right)\left(x\right)\right]-\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[{\kappa }_n{\lambda }_n\left(f\right)\left(x\right)\right]-\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right),\hfill \end{array}$$

where ${\displaystyle \varsigma \left(x\right)=\sum _{\ell =0}^3{\chi }_{\ell ,0}({\sigma }_{\ell },x_{\ell })}$ and ${\displaystyle {\kappa }_n=\sum _{\begin{array}{c}\ell =0\\ \ell \ne n\end{array}}^3{\chi }_{\ell ,0}({\sigma }_{\ell },x_{\ell })}$.

Then

$$\begin{array}{cc}\hfill & \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right):=\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\varsigma \left(x\right)\left(\sum _{k=0}^3{\lambda }_k\left(f\right)\left(x\right)\right)\right]-\sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\varsigma \left(x\right){\lambda }_k\left(f\right)\left(x\right)\right]\hfill \\ \hfill & -\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\sum _{k=0}^3{\kappa }_k{\lambda }_k\left(f\right)\left(x\right)\right]+\sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[{\kappa }_k{\lambda }_k\left(f\right)\left(x\right)\right]\hfill \\ \hfill & -\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right).\hfill \end{array}$$

As a consequence we have that

$$\begin{array}{cc}\hfill & \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right):=\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\varsigma \left(x\right)\left(\sum _{k=0}^3{\lambda }_k\left(f\right)\left(x\right)\right)-\sum _{k=0}^3{\kappa }_k{\lambda }_k\left(f\right)\left(x\right)\right]\hfill \\ \hfill & +\sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[{\kappa }_k{\lambda }_k\left(f\right)\left(x\right)-\varsigma \left(x\right){\lambda }_k\left(f\right)\left(x\right)\right]-\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right),\hfill \end{array}$$

i.e.,

$$\begin{array}{cc}\hfill \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right):=& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\sum _{k=0}^3\left({\chi }_{k,0}({\sigma }_k,x_k)\right){\lambda }_k\left(f\right)\left(x\right)\right]\hfill \\ \hfill & -\sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\left({\chi }_{k,0}({\sigma }_k,x_k)\right){\lambda }_k\left(f\right)\left(x\right)\right]\hfill \\ \hfill & -\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right)\hfill \end{array}$$

or equivalently

$$\begin{array}{cc}\hfill \psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right):=& \psi \mathcal{D}\circ {\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)-\sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[\left({\chi }_{k,0}({\sigma }_k,x_k)\right){\lambda }_k\left(f\right)\left(x\right)\right]\hfill \\ \hfill & -\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(x\right){\lambda }_n\left(f_m\right)\left(x\right).\hfill \end{array}$$

□

Notation $L_{n,m}$ and $h_n$ for all $n,m=0,1,2,3$, can be improved but we have decided to keep it at this level to make easier to write and read the following computations.

Definition 6.

For $\delta =({\delta }_0,{\delta }_1,{\delta }_2,{\delta }_3)\in {[0,1]}^4$, and $\mu =({\mu }_0,{\mu }_1,{\mu }_2,{\mu }_3)$ where ${\mu }_k({\zeta }_k,x_k)$ is a fractal measure for $k=0,1,2,3$ according to Definition 1. Denote ${\varkappa }_1=({\varkappa }_{0,1},{\varkappa }_{1,1},{\varkappa }_{2,1},{\varkappa }_{3,1})$, ${\varkappa }_0=({\varkappa }_{0,0},{\varkappa }_{1,0},{\varkappa }_{2,0},{\varkappa }_{3,0})$ and $\rho =({\rho }_0,{\rho }_1,{\rho }_2,{\rho }_3)\in {[0,1]}^4$ where ${\varkappa }_{k,1}({\rho }_k,x_k)$ and ${\varkappa }_{k,0}({\rho }_k,x_k)$ and are given by Definition 3 for $k=0,1,2,3$.

Given $g:\Omega \to \mathbb{H}$ such that ${\displaystyle \frac{{\partial }_{{\mu }_n}^{{\delta }_n}g\left(x\right)}{\partial {\left(x_n\right)}^{{\zeta }_n}}}$ there exists for all $x\in \Omega$ and all $n=0,1,2,3$. The quaternionic right ψ-proportional δ-fractal derivative of g with respect to μ and ρ, is given by

$$\begin{array}{cc}\hfill \psi {\mathcal{D}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(x\right):=& \sum _{n=0}^3{\displaystyle \frac{{\partial }_{{\mu }_n}^{{\rho }_n,{\delta }_n}g\left(x\right)}{\partial {\left(x_n\right)}^{{\zeta }_n}}}{\psi }_n\hfill \\ \hfill =& \sum _{n=0=m}^3{\psi }_m{\psi }_n\left({\varkappa }_{n,1}({\rho }_n,x_n)g_m\left(x\right)+{\varkappa }_{n,0}({\rho }_n,x_n){\displaystyle \frac{{\partial }_{{\mu }_n}^{{\delta }_n}{g}_m\left(x\right)}{\partial {\left(x_n\right)}^{{\zeta }_n}}}\right).\hfill \end{array}$$

Remark 3.

Consider $g:\Omega \to \mathbb{H}$ such that

$$\begin{array}{c}\hfill {\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\left(x\right)={\int }_0^{x_n}{\displaystyle \frac{{\partial }_{{\mu }_n}^{{\delta }_n}{g}_m\left(x\right)}{\partial {\left(t_n\right)}^{{\zeta }_n}}}dt,{\lambda }_{{\mu }_n}^{{\delta }_n}\left(g\right)\left(x\right)=\sum _{m=0}^3{\psi }_m{\int }_0^{x_n}{\displaystyle \frac{{\partial }_{{\mu }_n}^{{\delta }_n}{g}_m\left(x\right)}{\partial {\left(t_n\right)}^{{\zeta }_n}}}dt\end{array}$$

there exist for $n,m=0,1,2,3$. If ${\varkappa }_{n,0}({\rho }_n,x_n)\ne 0$ and ${\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\left(x\right)\ne 0$ for all $x=(x_0,x_1,x_2,x_3)\in \Omega$ and all $n=0,1,2,3$, then repeating several computations of the previous proof we can see that

$$\begin{array}{cc}\hfill \psi {\mathcal{D}}_r\circ {\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left(x\right):=& \psi {\mathcal{D}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(x\right)+{\mathcal{E}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(x\right)+\sum _{n=0=m}^3{\psi }_m{\psi }_n{T}_{n,m}\left[g\right]\left(x\right){\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\left(x\right),\hfill \end{array}$$

(5)

for all $x\in \Omega$, where

$$\begin{array}{cc}\hfill {\mathcal{E}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(x\right):=& \sum _{\begin{array}{c}n=0=k\\ n\ne k\end{array}}^3{\displaystyle \frac{\partial }{\partial x_n}}\left[\left({\varkappa }_{k,0}({\rho }_k,x_k)\right){\lambda }_{{\mu }_k}^{{\delta }_k}\left(g\right)\left(x\right)\right]{\psi }_n\hfill \\ \hfill {\mathfrak{L}}_{\mu }^{\rho ,\delta }\left(g\right)\left(x\right)=& \sum _{k=0}^3\left({\varkappa }_{k,0}({\rho }_k,x_k)\right){\lambda }_{{\nu }_k}^{{\delta }_k}\left(g\right)\left(x\right),\hfill \\ \hfill T_{n,m}\left[g\right]\left(x\right):=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\varkappa }_{n,0}({\rho }_n,x_n)}{e^{l_{n,m}\left(x\right)}}}\right)e^{l_{n,m}\left(x\right)},\hfill \\ \hfill l_{n,m}\left(x\right)=& {\int }_0^{x_n}{\displaystyle \frac{{\varkappa }_{n,1}({\rho }_n,t_n)}{{\varkappa }_{n,0}({\rho }_n,t_n)}}{\displaystyle \frac{g_m}{{\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)}}dt,\hfill \end{array}$$

for $n,m\in \{0,1,2,3\}$.

Assuming hypothesis and notations of Proposition 1 and Remark 3 let us present some consequences of quatertionic Borel-Pompeiu and Stokes formulas.

Proposition 2.

Let $\Omega \subset \mathbb{H}$ be a domain such that $\partial \Omega$ is a 3-dimensional smooth surface. If ${\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right],{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\in C^1(\Omega ,\mathbb{H})$ then

$$\begin{array}{c}{\int }_{\partial \Omega }(K_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(\tau \right)+{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left(\tau \right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x))\hfill \\ \hfill & -{\int }_{\Omega }\left(K_{\psi }(y-x)\psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(y\right)-\psi {\mathcal{D}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(y\right)K_{\psi }(y-x)\right)dy\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left({\mathcal{E}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(y\right)+\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(y\right){\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\left(y\right)\right)dy\hfill \\ \hfill & -{\int }_{\Omega }\left({\mathcal{E}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(y\right)+\sum _{n=0=m}^3{\psi }_m{\psi }_n{T}_{n,m}\left[g\right]\left(y\right){\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\left(y\right)\right)K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)+{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left(x\right),\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega },\hfill \end{array}\right.\hfill \end{array}$$

(6)

and

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]{\sigma }_x^{\psi }{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]={\int }_{\Omega }\left({\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\left[f\right]+\psi {\mathcal{D}}_{r,\mu }^{\rho ,\delta }\left[g\right]{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\right)dx+\hfill \\ \hfill & +{\int }_{\Omega }{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left({\mathcal{E}}_{\nu }^{\sigma ,\beta }\left[f\right]+\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]{\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\right)dx\hfill \\ \hfill & +{\int }_{\Omega }\left({\mathcal{E}}_{r,\mu }^{\rho ,\delta }\left[g\right]+\sum _{n=0=m}^3{\psi }_m{\psi }_n{T}_{n,m}\left[g\right]{\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\right){\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]dx.\hfill \end{array}$$

(7)

Proof. 

The formulas follow by application of quaternionic Borel-Pompieu and Stokes formula, functions ${\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]$, ${\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]$ and the usage of identities given in Proposition 1 and Remark 3. □

Remark 4.

In case in which ${\mathfrak{L}}_{\nu }^{\sigma ,\beta }$ and ${\mathfrak{L}}_{\mu }^{\rho ,\delta }$ are invertible operators we can improve formula (6) to obtain the quaternionc values of f and g. In addition, if $f\in \mathrm{Ker}\left(\psi {\mathcal{D}}_{\nu }^{\sigma ,\beta }\right)$ and $g\in \mathrm{Ker}\left(\psi {\mathcal{D}}_{r,\mu }^{\rho ,\delta }\right)$ then

$$\begin{array}{cc}& {\int }_{\partial \Omega }(K_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(\tau \right)+{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left(\tau \right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x))\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left({\mathcal{E}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(y\right)+\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]\left(y\right){\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\left(y\right)\right)dy\hfill \\ \hfill & -{\int }_{\Omega }\left({\mathcal{E}}_{r,\mu }^{\rho ,\delta }\left[g\right]\left(y\right)+\sum _{n=0=m}^3{\psi }_m{\psi }_n{T}_{n,m}\left[g\right]\left(y\right){\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\left(y\right)\right)K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]\left(x\right)+{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left(x\right),\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega },\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]{\sigma }_x^{\psi }{\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]={\int }_{\Omega }{\mathfrak{L}}_{\mu }^{\rho ,\delta }\left[g\right]\left({\mathcal{E}}_{\nu }^{\sigma ,\beta }\left[f\right]+\sum _{n=0=m}^3{\psi }_n{\psi }_m{L}_{n,m}\left[f\right]{\lambda }_{{\nu }_n}^{{\beta }_n}\left(f_m\right)\right)dx\hfill \\ \hfill & +{\int }_{\Omega }\left({\mathcal{E}}_{r,\mu }^{\rho ,\delta }\left[g\right]+\sum _{n=0=m}^3{\psi }_m{\psi }_n{T}_{n,m}\left[g\right]{\lambda }_{{\mu }_n}^{{\delta }_n}\left(g_m\right)\right){\mathfrak{L}}_{\nu }^{\sigma ,\beta }\left[f\right]dx.\hfill \end{array}$$

4. Quaternionic $\beta$-Proportional Fractal Fueter Operator with Truncated Exponential Fractal Measure

From now on, partial differential operators given by Remarks 1 and 2 are considered, and let $k:=(k_0,k_1,k_2,k_3)\in {\mathbb{N}}^4$, $\sigma =({\sigma }_0,{\sigma }_1,{\sigma }_2,{\sigma }_3),\beta =({\beta }_0,{\beta }_1,{\beta }_2,{\beta }_3)\in {[0,1]}^4$, $\alpha =({\alpha }_0,{\alpha }_1,{\alpha }_2,{\alpha }_3)\in (0,1]$ and for $n=0,1,2,3$.

Let $\Omega \subset \mathbb{H}$ be a domain and $f\in C^1(\Omega ,\mathbb{R})$. We will use the proportional ${\beta }_n$-fractal partial derivatives

$${\displaystyle \frac{{\partial }^{{\sigma }_n,{\beta }_n}f}{\partial x_{{\alpha }_n,k_n}}}\left(x\right):=(1-{\sigma }_n)f\left(x\right)+{\sigma }_n{\displaystyle \frac{{\displaystyle \frac{\partial f^{{\beta }_n}}{\partial x_n}}\left(x\right)}{{\displaystyle \frac{\partial e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}{\partial x_n}}}},$$

for all $x={\sum }_{n=0}^3\psi x_n\in \Omega$.

Definition 7.

Let $\Omega \subset \mathbb{H}$ be a domain. Given $f={\sum }_{\ell =0}^3{\psi }_{\ell }{f}_{\ell }\in C^1(\Omega ,\mathbb{H})$, where $f_0,f_1,f_2,f_3$ are real valued functions. Define

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{\alpha ,k}^{\sigma ,\beta }f\right)\left(x\right):=& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{\displaystyle \frac{{\partial }^{{\sigma }_n,{\beta }_n}{f}_{\ell }}{\partial x_{{\alpha }_n,k_n}}}\left(x\right)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill =& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }\left((1-{\sigma }_n)f_{\ell }\left(x\right)+{\sigma }_n{\displaystyle \frac{{\displaystyle \frac{\partial f_{\ell }^{{\beta }_n}}{\partial x_n}}\left(x\right)}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill H_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right):=& {\int }_0^{x_n}{\displaystyle \frac{{\sigma }_n-1}{{\sigma }_n}}\left({\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}\right)f_{\ell }{\left(x\right)}^{1-{\beta }_n}d{x}_n\hfill \end{array}$$

(10)

and to simplify the notation in the proof of the next statement use $h_{n,\ell }\left(x\right)=H_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left(f_{\ell }\right)\left(x\right)$ for all $n,\ell =0,1,2,3$. In addition,

$$\begin{array}{cc}\hfill T_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right):=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right)e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n},\hfill \\ \hfill \psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right),\hfill \end{array}$$

where ${\displaystyle I^{{\beta }_n}\left[f\right]\left(x\right)=\sum _{\ell =0}^3{\psi }_{\ell }{f}_{\ell }{\left(x\right)}^{{\beta }_n}}$ for all $x\in \Omega$ and $n,\ell =0,1,2,3$.

Proposition 3.

Given $f={\sum }_{\ell =0}^3{\psi }_{\ell }{f}_{\ell }\in C^1(\Omega ,\mathbb{H})$. Then

$$\begin{array}{cc}\hfill & \psi \mathcal{D}\left[\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\right]\left(x\right)\hfill \\ \hfill =& \left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(x\right)+\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right),\hfill \end{array}$$

(11)

for all $x\in \Omega$.

Proof. 

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,\ell }\left(x_n\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)=& \left[{\displaystyle \frac{1-{\sigma }_n}{{\sigma }_n}}\left({\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}\right){\left(f_{\ell }\left(x\right)\right)}^{1-{\beta }_n}{f}_{\ell }^{{\beta }_n}\left(x\right)+{\displaystyle \frac{\partial f_{\ell }^{{\beta }_n}}{\partial x_n}}\left(x\right)\right]e^{h_{n,\ell }\left(x\right)}\hfill \\ \hfill =& \left[(1-{\sigma }_n)f_{\ell }\left(x\right)+{\sigma }_n{\displaystyle \frac{{\displaystyle \frac{\partial f_{\ell }^{{\beta }_n}}{\partial x_n}}\left(x\right)}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right]{\displaystyle \frac{1}{{\sigma }_n}}{e}^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n},\hfill \\ \hfill {\displaystyle \frac{{\partial }^{{\sigma }_n,{\beta }_n}}{\partial x_{{\alpha }_n,k_n}}}{f}_{\ell }\left(x\right)=& {\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {(}^{\psi }{\mathcal{D}}_{\alpha ,k}^{\sigma ,\beta }f)\left(x\right)=& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{{\partial }^{{\sigma }_n,{\beta }_n}}{\partial x_{{\alpha }_n,k_n}}}f\left(x\right)\hfill \\ \hfill =& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right).\hfill \end{array}$$

The identities

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)={\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{e}^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)\hfill \\ \hfill =& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right)e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}+{\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{\displaystyle \frac{\partial }{\partial x_n}}\left(e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)\hfill \\ \hfill =& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)-{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right)e^{h_{n,\ell }\left(x\right)}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\hfill \\ \hfill =& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)-T_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)\left(x\right)\hfill \end{array}$$

imply that

$$\begin{array}{cc}\hfill {(}^{\psi }{\mathcal{D}}_{\alpha ,k}^{\sigma ,\beta }f)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }\left[{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right)-T_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)\right]\hfill \\ \hfill =& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{\displaystyle \frac{\partial }{\partial x_n}}\left[{\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right]-\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)\hfill \\ \hfill =& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[{\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\sum _{\ell =0}^3{\psi }_{\ell }{f}_{\ell }{\left(x\right)}^{{\beta }_n}\right]-\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)\hfill \\ \hfill =& \sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left[{\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{I}^{{\beta }_n}\left[f\right]\left(x\right)\right]-\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right).\hfill \end{array}$$

For each $n=0,1,2,3$ we see that

$${\displaystyle \frac{{\sigma }_n}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}{I}^{{\beta }_n}\left[f\right]\left(x\right)=\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right)-\sum _{\begin{array}{c}\ell =0\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right).$$

Therefore,

$$\begin{array}{cc}\hfill & \psi \mathcal{D}\left[\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\right]\left(x\right)\hfill \\ \hfill =& \left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(x\right)+\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)+\sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right).\hfill \end{array}$$

□

Remark 5.

Denote $v=(v_0,v_1,v_2,v_3)\in {\mathbb{N}}^4$, $\rho =({\rho }_0,{\rho }_1,{\rho }_2,{\rho }_3),\delta =({\delta }_0,{\delta }_1,{\delta }_2,{\delta }_3)\in {(0,1]}^4$, $\gamma =({\gamma }_0,{\gamma }_1,{\gamma }_2,{\gamma }_3)\in {(0,1]}^4$ and for $n=0,1,2,3$. We will use the the proportional ${\delta }_n$-fractal partial derivative ${\displaystyle \frac{{\partial }^{{\rho }_n,{\delta }_n}}{\partial x_{{\gamma }_n,k_n}}}$. Recall that if $g\in C^1(\Omega ,\mathbb{R})$ then

$${\displaystyle \frac{{\partial }^{{\rho }_n,{\delta }_n}g}{\partial x_{{\gamma }_n,k_n}}}\left(x\right):=(1-{\rho }_n)g\left(x\right)+{\rho }_n{\displaystyle \frac{{\displaystyle \frac{\partial g^{{\delta }_n}}{\partial x_n}}\left(x\right)}{{\displaystyle \frac{\partial e{\left(x_n^{{\gamma }_n}\right)}_{k_n}}{\partial x_n}}}},$$

for all $x={\sum }_{n=0}^3\psi x_n\in \Omega$.

If $g={\sum }_{\ell =0}^3{\psi }_{\ell }{g}_{\ell }\in C^1(\Omega ,\mathbb{H})$, where $g_0,g_1,g_2,g_3$ are real valued functions. Define the right version of the operator given by (8) as follows:

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right):=& \sum _{n=0=\ell }^3{\psi }_{\ell }{\displaystyle \frac{{\partial }^{{\rho }_n,{\delta }_n}{g}_{\ell }}{\partial x_{{\gamma }_n,m_n}}}\left(x\right){\psi }_n,\hfill \\ \hfill H_{{\gamma }_n,k_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(x\right):=& {\int }_0^{x_n}{\displaystyle \frac{{\rho }_n-1}{{\rho }_n}}\left({\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\gamma }_n}\right)}_{m_n}\right)g_{\ell }{\left(x\right)}^{1-{\delta }_n}d{x}_n.\hfill \end{array}$$

and use $j_{n,\ell }\left(x\right)=H_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left(g_{\ell }\right)\left(x\right)$ for all $n,\ell =0,1,2,3$. Denote

$$\begin{array}{cc}\hfill S_{{\gamma }_n,m_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(x\right):=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_{n,\ell }\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\gamma }_n}\right)}_{m_n}}}\right)e^{j_{n,\ell }\left(x\right)}{g}_{\ell }{\left(x\right)}^{{\delta }_n},\hfill \\ \hfill \psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right):=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{\displaystyle \frac{\partial }{\partial x_n}}{I}^{{\delta }_{\ell }}\left[g\right]\left(x\right){\psi }_n.\hfill \end{array}$$

From similar computations to presented in the previous proof we can obtain the right version of (11):

$$\begin{array}{cc}\hfill & \psi {\mathcal{D}}_r\left[\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{I}^{{\delta }_{\ell }}\left[g\right]\right]\left(x\right)\hfill \\ \hfill =& \left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)+\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,m_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(x\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right),\hfill \end{array}$$

(12)

for all $x\in \Omega$

Corollary 1.

Let $\Omega \subset \mathbb{H}$ be a domain such that $\partial \Omega$ is a 3-dimensional smooth surface. In agreement with notation in Definition 7 and Remark 5 we have:

(1)

If $\beta =(1,1,1,1)$ then operators given in Definition 7 are represented as follows:

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{\alpha ,k}^{\sigma ,\beta }f\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }\left((1-{\sigma }_n)f_{\ell }\left(x\right)+{\sigma }_n{\displaystyle \frac{{\displaystyle \frac{\partial f_{\ell }}{\partial x_n}}\left(x\right)}{{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right),\hfill \\ \hfill h_n\left(x\right)=H_{{\alpha }_n,k_n}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\sigma }_n-1}{{\sigma }_n}}\left[e{\left(x_n^{{\alpha }_n}\right)}_{k_n}-1\right],\hfill \\ \hfill T_{{\alpha }_n,k_n}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_n\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right)e^{h_n\left(x\right)}{f}_{\ell }\left(x\right),\hfill \\ \hfill I^1\left[f\right]=& f\hfill \\ \hfill \psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}f\left(x\right),\hfill \end{array}$$

for all $x\in \Omega$ and (11) becomes at

$$\begin{array}{cc}\hfill \psi \mathcal{D}\left[\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}f\right]\left(x\right)=& \left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(x\right)+A\left(x\right)f\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right),\hfill \end{array}$$

where

$$A\left(x\right):=\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_n\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\alpha }_n}\right)}_{k_n}}}\right)e^{h_n\left(x\right)},$$

for all $x\in \Omega$. Another important cases are the following:

(a)

If $\beta =(1,1,1,1)$ and $k=(1,1,1,1)$ then

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{\alpha ,k}^{\sigma ,\beta }f\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }\left((1-{\sigma }_n)f_{\ell }\left(x\right)+{\sigma }_n{\displaystyle \frac{{\displaystyle \frac{\partial f_{\ell }}{\partial x_n}}\left(x\right)}{{\alpha }_n{x}_n^{{\alpha }_n-1}}}\right),\hfill \\ \hfill h_n\left(x\right)=H_{{\alpha }_n,1}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\sigma }_n-1}{{\sigma }_n}}{x}_n^{{\alpha }_n},\hfill \\ \hfill T_{{\alpha }_n,1}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_n\left(x\right)}{\alpha }_n{x}_n^{{\alpha }_n-1}}}\right)e^{h_n\left(x\right)}{f}_{\ell }\left(x\right),\hfill \\ \hfill I^1\left[f\right]=& f\hfill \\ \hfill \psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}}}{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}f\left(x\right),\hfill \end{array}$$

for all $x\in \Omega$ and (11) becomes at

$$\begin{array}{cc}\hfill \psi \mathcal{D}\left[\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}}}f\right]\left(x\right)=& \left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(x\right)+A\left(x\right)f\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right),\hfill \end{array}$$

where

$$A\left(x\right):=\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_n\left(x\right)}{\alpha }_n{x}_n^{{\alpha }_n-1}}}\right)e^{h_n\left(x\right)},$$

for all $x\in \Omega$.

(b)

If $\beta =(1,1,1,1)$ and $k=(\infty ,\infty ,\infty ,\infty )$ then

$$\begin{array}{cc}\hfill \left({\mathcal{D}}_{\alpha ,k}^{\sigma ,\beta }f\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }\left((1-{\sigma }_n)f_{\ell }\left(x\right)+{\sigma }_n{\displaystyle \frac{{\displaystyle \frac{\partial f_{\ell }}{\partial x_n}}\left(x\right)}{{\alpha }_n{x}_n^{{\alpha }_n-1}{e}^{x_n^{{\alpha }_n}}}}\right),\hfill \\ \hfill h_n\left(x\right)=H_{{\alpha }_n,\infty }^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\sigma }_n-1}{{\sigma }_n}}\left[e^{x_n^{{\alpha }_n}}-1\right],\hfill \\ \hfill T_{{\alpha }_n,\infty }^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_n\left(x\right)}{\alpha }_n{x}_n^{{\alpha }_n-1}{e}^{x_n^{{\alpha }_n}}}}\right)e^{h_n\left(x\right)}{f}_{\ell }\left(x\right),\hfill \\ \hfill I^1\left[f\right]=& f\hfill \\ \hfill \psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}{e}^{x_{\ell }^{{\alpha }_{\ell }}}}}{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}f\left(x\right),\hfill \end{array}$$

for all $x\in \Omega$ and (11) becomes at

$$\begin{array}{cc}\hfill \psi \mathcal{D}\left[\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}{e}^{x_{\ell }^{{\alpha }_{\ell }}}}}f\right]\left(x\right)=& \left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(x\right)+A\left(x\right)f\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right),\hfill \end{array}$$

where

$$A\left(x\right):=\sum _{n=0}^3{\psi }_n{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\sigma }_n}{e^{h_n\left(x\right)}{\alpha }_n{x}_n^{{\alpha }_n-1}{e}^{x_n^{{\alpha }_n}}}}\right)e^{h_n\left(x\right)},$$

for all $x\in \Omega$.

(2)

If $\delta =(1,1,1,1)$ then the operators given in Remark 5 are represented by

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_{\ell }{\displaystyle \frac{{\partial }^{{\rho }_n,1}{g}_{\ell }}{\partial x_{{\gamma }_n,m_n}}}\left(x\right){\psi }_n,\hfill \\ \hfill j_n\left(x\right)=H_{{\gamma }_n,k_n}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\rho }_n-1}{{\rho }_n}}\left[e{\left(x_n^{{\gamma }_n}\right)}_{m_n}-1\right],\hfill \\ \hfill S_{{\gamma }_n,m_n}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\gamma }_n}\right)}_{m_n}}}\right)e^{j_n\left(x\right)}{g}_{\ell }\left(x\right),\hfill \\ \hfill \psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{\displaystyle \frac{\partial }{\partial x_n}}g\left(x\right){\psi }_n\hfill \end{array}$$

and identity (12) is

$$\begin{array}{c}\hfill \psi {\mathcal{D}}_r\left[\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}g\right]\left(x\right)=\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)+g\left(x\right)B\left(x\right)+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right),\end{array}$$

where

$$B\left(x\right)=\sum _{n=0}^3{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\gamma }_n}\right)}_{m_n}}}\right)e^{j_n\left(x\right)}{\psi }_n,$$

for all $x\in \Omega$.

(a)

If $\delta =(1,1,1,1)$ then the operators given in Remark 5 are represented by

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_{\ell }{\displaystyle \frac{{\partial }^{{\rho }_n,1}{g}_{\ell }}{\partial x_{{\gamma }_n,m_n}}}\left(x\right){\psi }_n,\hfill \\ \hfill j_n\left(x\right)=H_{{\gamma }_n,k_n}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\rho }_n-1}{{\rho }_n}}\left[e{\left(x_n^{{\gamma }_n}\right)}_{m_n}-1\right],\hfill \\ \hfill S_{{\gamma }_n,m_n}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\gamma }_n}\right)}_{m_n}}}\right)e^{j_n\left(x\right)}{g}_{\ell }\left(x\right),\hfill \\ \hfill \psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{\displaystyle \frac{\partial }{\partial x_n}}g\left(x\right){\psi }_n\hfill \end{array}$$

and identity (12) is

$$\begin{array}{c}\hfill \psi {\mathcal{D}}_r\left[\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}g\right]\left(x\right)=\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)+g\left(x\right)B\left(x\right)+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right),\end{array}$$

where

$$B\left(x\right)=\sum _{n=0}^3{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\displaystyle \frac{d}{d{x}_n}}e{\left(x_n^{{\gamma }_n}\right)}_{m_n}}}\right)e^{j_n\left(x\right)}{\psi }_n,$$

for all $x\in \Omega$.

(b)

If $\delta =(1,1,1,1)$ and $m=(1,1,1,1)$ then the operators given in Remark 5 are represented by

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_{\ell }{\displaystyle \frac{{\partial }^{{\rho }_n,1}{g}_{\ell }}{\partial x_{{\gamma }_n,1}}}\left(x\right){\psi }_n,\hfill \\ \hfill j_n\left(x\right)=H_{{\gamma }_n,k_n}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\rho }_n-1}{{\rho }_n}}{x}_n^{{\gamma }_n},\hfill \\ \hfill S_{{\gamma }_n,1}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\gamma }_n{x}_n^{{\gamma }_n-1}}}\right)e^{j_n\left(x\right)}{g}_{\ell }\left(x\right),\hfill \\ \hfill \psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}}}{\displaystyle \frac{\partial }{\partial x_n}}g\left(x\right){\psi }_n\hfill \end{array}$$

and identity (12) is

$$\begin{array}{c}\hfill \psi {\mathcal{D}}_r\left[\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}}}g\right]\left(x\right)=\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)+g\left(x\right)B\left(x\right)+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right),\end{array}$$

where

$$B\left(x\right)=\sum _{n=0}^3{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\gamma }_n{x}_n^{{\gamma }_n-1}}}\right)e^{j_n\left(x\right)}{\psi }_n,$$

for all $x\in \Omega$.

(c)

If $\delta =(1,1,1,1)$ and $m=(\infty ,\infty ,\infty ,\infty )$ then the operators given in Remark 5 are represented by

$$\begin{array}{cc}\hfill \left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)=& \sum _{n=0=\ell }^3{\psi }_{\ell }{\displaystyle \frac{{\partial }^{{\rho }_n,1}{g}_{\ell }}{\partial x_{{\gamma }_n,\infty }}}\left(x\right){\psi }_n,\hfill \\ \hfill j_n\left(x\right)=H_{{\gamma }_n,k_n}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{{\rho }_n-1}{{\rho }_n}}\left[e^{x_n^{{\gamma }_n}}-1\right],\hfill \\ \hfill S_{{\gamma }_n,\infty }^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right)=& {\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\gamma }_n{x}_n^{{\gamma }_n-1}{e}^{x_n^{{\gamma }_n}}}}\right)e^{j_n\left(x\right)}{g}_{\ell }\left(x\right),\hfill \\ \hfill \psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)=& \sum _{\begin{array}{c}\ell =0=n\\ \ell \ne n\end{array}}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}{e}^{x_{\ell }^{{\gamma }_{\ell }}}}}{\displaystyle \frac{\partial }{\partial x_n}}g\left(x\right){\psi }_n\hfill \end{array}$$

and identity (12) is

$$\begin{array}{c}\hfill \psi {\mathcal{D}}_r\left[\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}{e}^{x_{\ell }^{{\gamma }_{\ell }}}}}g\right]\left(x\right)=\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)+g\left(x\right)B\left(x\right)+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right),\end{array}$$

where

$$B\left(x\right)=\sum _{n=0}^3{\displaystyle \frac{\partial }{\partial x_n}}\left({\displaystyle \frac{{\rho }_n}{e^{j_n\left(x\right)}{\gamma }_n{x}_n^{{\gamma }_n-1}{e}^{x_n^{{\gamma }_n}}}}\right)e^{j_n\left(x\right)}{\psi }_n,$$

for all $x\in \Omega$.

Proposition 4.

Let $\Omega \subset \mathbb{H}$ be a domain such that $\partial \Omega$ is a 3-dimensional smooth surface. In agreement with notation in Definition 7 and Remark 5 let $f={\sum }_{\ell =0}^3{\psi }_{\ell }{f}_{\ell },g={\sum }_{\ell =0}^3{\psi }_{\ell }{g}_{\ell }\in C^1(\Omega ,\mathbb{H})$, where $f_{\ell },g_{\ell }$ are real valued functions. Then

$$\begin{array}{c}{\int }_{\partial \Omega }{K}_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{\frac{d}{d{\tau }_{\ell }}e{\left({\tau }_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\left(\tau \right)\right)\hfill \\ \hfill & +{\int }_{\partial \Omega }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{\frac{d}{d{\tau }_{\ell }}e{\left({\tau }_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{I}^{{\delta }_{\ell }}\left[g\right]\left(\tau \right)\right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x)\hfill \\ \hfill & -{\int }_{\Omega }\left[K_{\psi }(y-x)\left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(y\right)+\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(y\right)K_{\psi }(y-x)\right]dy\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(y\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(y\right)\right]dy\hfill \\ \hfill & -{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,m_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(y\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(y\right)\right]K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\displaystyle \sum _{\ell =0}^3\frac{{\sigma }_{\ell }}{\frac{d}{d{x}_{\ell }}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right)+\sum _{\ell =0}^3\frac{{\rho }_{\ell }}{\frac{d}{d{x}_{\ell }}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}{I}^{{\delta }_{\ell }}\left[g\right]\left(x\right),}\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega }.\hfill \end{array}\right.\hfill \end{array}$$

(13)

In addition,

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{I}^{{\delta }_{\ell }}\left[g\right]\right){\sigma }_x^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right)\right)\hfill \\ \hfill =& {\int }_{\Omega }\left(g\left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }f\right)\left(x\right)+\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }g\right)\left(x\right)f(x\right)dx\hfill \\ \hfill & +{\int }_{\Omega }g\left(x\right)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)\right]dx\hfill \\ \hfill & +{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,m_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(x\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)\right]f\left(x\right)dx\hfill \end{array}$$

(14)

Proof. 

It is a direct consequence of Definition 7 and Remark 5 using functions

${\displaystyle \sum _{\ell =0}^3\frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right)}$ and ${\displaystyle \sum _{\ell =0}^3\frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}{I}^{{\delta }_{\ell }}\left[g\right]\left(x\right)}$ and identities (11) and (12) in formulas (2) and (3). □

Remark 6.

In formulas (13) and (14), the operators $\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }$ and $\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }$ reflect the phenomenon of duality in quaternionic analysis due to the non-commutativity of quaterinonic algebra.

Corollary 2.

Let $\Omega \subset \mathbb{H}$ be a domain such that $\partial \Omega$ is a 3-dimensional smooth surface. In agreement with notation in Definition 7 and Remark 5 let $f={\sum }_{\ell =0}^3{\psi }_{\ell }{f}_{\ell },g={\sum }_{\ell =0}^3{\psi }_{\ell }{g}_{\ell }\in C^1(\Omega ,\mathbb{H})$, where $f_{\ell },g_{\ell }$ are real valued functions. Suppose that $f\in \mathrm{Ker}\left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }\right)$ and $g\in \mathrm{Ker}\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }\right)$. Then

$$\begin{array}{cc}& {\int }_{\partial \Omega }{K}_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{\tau }_{\ell }}}e{\left({\tau }_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\left(\tau \right)\right)\hfill \\ \hfill & +{\int }_{\partial \Omega }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{\tau }_{\ell }}}e{\left({\tau }_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{I}^{{\delta }_{\ell }}\left[g\right]\left(\tau \right)\right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x)\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(y\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(y\right)\right]dy\hfill \\ \hfill & -{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,m_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(y\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(y\right)\right]K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\displaystyle \sum _{\ell =0}^3\frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right)+\sum _{\ell =0}^3\frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}{I}^{{\delta }_{\ell }}\left[g\right]\left(x\right),}\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega }\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\gamma }_{\ell }}\right)}_{m_{\ell }}}}{I}^{{\delta }_{\ell }}\left[g\right]\right){\sigma }_x^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\displaystyle \frac{d}{d{x}_{\ell }}}e{\left(x_{\ell }^{{\alpha }_{\ell }}\right)}_{k_{\ell }}}}{I}^{{\beta }_{\ell }}\left[f\right]\left(x\right)\right)\hfill \\ \hfill =& {\int }_{\Omega }g\left(x\right)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,k_n}^{{\sigma }_n,{\beta }_n}\left[f_{\ell }\right]\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)\right]dx\hfill \\ \hfill & +{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,m_n}^{{\rho }_n,{\delta }_n}\left[g_{\ell }\right]\left(x\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)\right]f\left(x\right)dx\hfill \end{array}$$

Corollary 3.

Let $\Omega \subset \mathbb{H}$ be a domain such that $\partial \Omega$ is a 3-dimensional smooth surface. In agreement with notation in Definition 7 and Remark 5 let $f={\sum }_{\ell =0}^3{\psi }_{\ell }{f}_{\ell },g={\sum }_{\ell =0}^3{\psi }_{\ell }{g}_{\ell }\in C^1(\Omega ,\mathbb{H})$, where $f_{\ell },g_{\ell }$ are real valued functions. Suppose that $f\in \mathrm{Ker}\left(\psi {\mathcal{D}}_{\alpha ,k}^{\rho ,\beta }\right)$ and $g\in \mathrm{Ker}\left(\psi {\mathcal{D}}_{r,\gamma ,m}^{\rho ,\delta }\right)$. For fix $\beta =(1,1,1,1)$ and $\delta =(1,1,1,1)$ we have:

(1)

If $k=(1,1,1,1)$ and $m=(1,1,1,1)$, then

$$\begin{array}{cc}& {\int }_{\partial \Omega }{K}_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{\tau }_{\ell }^{{\alpha }_{\ell }-1}}}\right)f\left(\tau \right)+{\int }_{\partial \Omega }g\left(\tau \right)\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{\tau }_{\ell }^{{\gamma }_{\ell }-1}}}\right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x)\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,1}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(y\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(y\right)\right]dy\hfill \\ \hfill & -{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,1}^{{\rho }_n,1}\left[g_{\ell }\right]\left(y\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(y\right)\right]K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\displaystyle f\left(x\right)\sum _{\ell =0}^3\frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}}+g\left(x\right)\sum _{\ell =0}^3\frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}},}\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega }\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }g\left(x\right)\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}}}\right){\sigma }_x^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}}}\right)f\left(x\right)\hfill \\ \hfill =& {\int }_{\Omega }g\left(x\right)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,1}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)\right]dx\hfill \\ \hfill & +{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,1}^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)\right]f\left(x\right)dx,\hfill \end{array}$$

where operators $T_{{\alpha }_n,1}^{{\sigma }_n,1}$, $\psi W_{\alpha ,k}^{\sigma ,\beta }$, $S_{{\gamma }_n,1}^{{\rho }_n,1}$ and $\psi V_{\gamma ,m}^{\rho ,\delta }$ are represented in Corollary 1.

(2)

If $k=(\infty ,\infty ,\infty ,\infty )$ and $m=(\infty ,\infty ,\infty ,\infty )$, then

$$\begin{array}{cc}& {\int }_{\partial \Omega }{K}_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{\tau }_{\ell }^{{\alpha }_{\ell }-1}{e}^{{\tau }_{\ell }^{{\alpha }_{\ell }}}}}\right)f\left(\tau \right)+{\int }_{\partial \Omega }g\left(\tau \right)\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{\tau }_{\ell }^{{\gamma }_{\ell }-1}{e}^{{\tau }_{\ell }^{{\gamma }_{\ell }}}}}\right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x)\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,\infty }^{{\sigma }_n,1}\left[f_{\ell }\right]\left(y\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(y\right)\right]dy\hfill \\ \hfill & -{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,\infty }^{{\rho }_n,1}\left[g_{\ell }\right]\left(y\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(y\right)\right]K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\displaystyle f\left(x\right)\sum _{\ell =0}^3\frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}{e}^{x_{\ell }^{{\alpha }_{\ell }}}}+g\left(x\right)\sum _{\ell =0}^3\frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}{e}^{x_{\ell }^{{\gamma }_{\ell }}}},}\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega }\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }g\left(x\right)\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}{e}^{x_{\ell }^{{\gamma }_{\ell }}}}}\right){\sigma }_x^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}{e}^{x_{\ell }^{{\alpha }_{\ell }}}}}\right)f\left(x\right)\hfill \\ \hfill =& {\int }_{\Omega }g\left(x\right)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,\infty }^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)\right]dx\hfill \\ \hfill & +{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,\infty }^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)\right]f\left(x\right)dx,\hfill \end{array}$$

where $T_{{\alpha }_n,\infty }^{{\sigma }_n,1}$, $\psi W_{\alpha ,k}^{\sigma ,\beta }$, $S_{{\gamma }_n,\infty }^{{\rho }_n,1}$ and $\psi V_{\gamma ,m}^{\rho ,\delta }$ are given in Corollary 1.

(3)

If $k=(1,1,1,1)$ and $m=(\infty ,\infty ,\infty ,\infty )$, then

$$\begin{array}{cc}& {\int }_{\partial \Omega }{K}_{\psi }(\tau -x){\sigma }_{\tau }^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{\tau }_{\ell }^{{\alpha }_{\ell }-1}}}\right)f\left(\tau \right)+{\int }_{\partial \Omega }g\left(\tau \right)\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{\tau }^{{\gamma }_{\ell }-1}{e}^{{\tau }_{\ell }^{{\gamma }_{\ell }}}}}\right){\sigma }_{\tau }^{\psi }{K}_{\psi }(\tau -x)\hfill \\ \hfill & -{\int }_{\Omega }{K}_{\psi }(y-x)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,1}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(y\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(y\right)\right]dy\hfill \\ \hfill & -{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,\infty }^{{\rho }_n,1}\left[g_{\ell }\right]\left(y\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(y\right)\right]K_{\psi }(y-x)dy\hfill \\ \hfill =& \left\{\begin{array}{cc}{\displaystyle f\left(x\right)\sum _{\ell =0}^3\frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}}+g\left(x\right)\sum _{\ell =0}^3\frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}{e}^{x_{\ell }^{{\gamma }_{\ell }}}},}\hfill & x\in \Omega ,\hfill \\ 0,\hfill & x\in \mathbb{H}\setminus \overline{\Omega }\hfill \end{array}\right.\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\int }_{\partial \Omega }g\left(x\right)\left(\sum _{\ell =0}^3{\displaystyle \frac{{\rho }_{\ell }}{{\gamma }_{\ell }{x}_{\ell }^{{\gamma }_{\ell }-1}{e}^{x_{\ell }^{{\gamma }_{\ell }}}}}\right){\sigma }_x^{\psi }\left(\sum _{\ell =0}^3{\displaystyle \frac{{\sigma }_{\ell }}{{\alpha }_{\ell }{x}_{\ell }^{{\alpha }_{\ell }-1}}}\right)f\left(x\right)\hfill \\ \hfill =& {\int }_{\Omega }g\left(x\right)\left[\sum _{n=0=\ell }^3{\psi }_n{\psi }_{\ell }{T}_{{\alpha }_n,1}^{{\sigma }_n,1}\left[f_{\ell }\right]\left(x\right)+\psi W_{\alpha ,k}^{\sigma ,\beta }\left[f\right]\left(x\right)\right]dx\hfill \\ \hfill & +{\int }_{\Omega }\left[\sum _{n=0=\ell }^3{\psi }_{\ell }{S}_{{\gamma }_n,\infty }^{{\rho }_n,1}\left[g_{\ell }\right]\left(x\right){\psi }_n+\psi V_{\gamma ,m}^{\rho ,\delta }\left[g\right]\left(x\right)\right]f\left(x\right)dx,\hfill \end{array}$$

where operators $T_{{\alpha }_n,1}^{{\sigma }_n,1}$, $\psi W_{\alpha ,k}^{\sigma ,\beta }$, $S_{{\gamma }_n,\infty }^{{\rho }_n,1}$ and $\psi V_{\gamma ,m}^{\rho ,\delta }$ are given in Corollary 1.

(4)

For $k=(\infty ,\infty ,\infty ,\infty )$ and $m=(1,1,1,1)$ a similar result is in fact true.

5. Discussion

This paper establishes the foundations of a quaternionic function theory associated to a proportional and fractional-fractal $\psi$-Fueter operator associated to a fractal measure. Also this work extends the quaternionic hiperholomorphic function theory. So what other results can be extended to this recent function theory?