Fundamental Structure of General Stochastic Dynamical Systems: High-Dimension Case

No one has proved that mathematically general stochastic dynamical systems have a special structure. Thus, we introduce a structure of a general stochastic dynamical system. According to scientific understanding, we assert that its deterministic part can be decomposed into three significant parts: the gradient of the potential function, friction matrix and Lorenz matrix. Our previous work proved this structure for the low-dimension case. In this paper, we prove this structure for the high-dimension case. Hence, this structure of general stochastic dynamical systems is fundamental.


Introduction
Stochastic di erential equations are widely used to describe random phenomena in complex systems in physics, biology, and chemistry. For such a stochastic dynamical system, researchers usually build an appropriate mathematical model based on basic scienti c laws and analyze or simulate it to gain insights about its complex phenomena. However, these models are proposed to solve speci c scienti c problems [1][2][3]. Until now, the general theory for stochastic di erential equations has been limited.
To gain a deeper understanding of the dynamic behaviors of general stochastic dynamical systems requires the exploration of their intrinsic mechanisms. In 2005, Ao [4] proposed Ao decomposition, which demonstrates that the deterministic part of general stochastic dynamical systems can be decomposed into three signi cant parts: the friction force, gradient of the potential function, and Lorenz force.
is inspired much work [5,6]. We discuss the scienti c signi cance of these three terms.

Potential Function.
From the biological point of view, the potential function can be explained by evolution theory. As we know, the fundamental nature of biology is determined by evolution. To explain adaptation and speciation, Darwin [7] formulated the theory of evolution based on natural selection. Accordingly, Fisher [8] proposed the fundamental theorem of natural selection, indicating that the increase rate of mean tness by natural selection is equal to its genetic variance in tness. In 1932, Wright [9] proposed the tness landscape concept, by which evolutionary adaptation may be seen as a hill-climbing process on the mean tness landscape until a local mean tness peak is reached. In 1940, Waddington [10] proposed the developmental landscape, which is equivalent to the tness landscape. Wright's tness landscape and Fisher's fundamental theorem of natural selection have been widely used to interpret adaptation as mean tness maximization (see Figure 1).
is phenomenon is often illustrated on mathematical landscapes as balls rolling downhill. A ball experiencing gravity tends to a minimum of the gravitational potential energy p(x) as a function of its spatial position x. e force on the ball is given by the local slope, f � − (dp(x)/dx).
e potential landscape is also called a potential function or energy function, which has been applied in fields such as physics, biology, and chemistry [11,12]. It can compare the relative stability of different attractors [13], account for the transition rates between neighboring steady states induced by noise [14], and provide an intuitive picture that reveals the essential mechanism underlying the complex system [15]. In physics, the potential function is closely related to the non-equilibrium thermodynamic framework [16]; in chemistry, it provides useful explanations for protein folding [17,18]; in biology, it has been used to explore basic problems in evolution such as the robustness, adaptability, and efficiency of real biological networks [19]. Until now, the general existence of the potential function remains unsolved. Researchers such as Prigogine et al. [20][21][22] have insisted that the potential function does not exist in non-equilibrium systems because they have not found it.

e Friction Matrix.
e friction matrix (the frictional force) represents dissipation. In this case, the energy of a dynamical system decreases, so the potential function decreases and the corresponding fitness increases. A system that has only friction is a gradient system (see green thick arrow in Figure 1).

1.3.
e Lorenz Matrix. Interestingly, Wright's fitness landscape theory [9] cannot explain the Red Queen hypothesis proposed by Valen [23], which illustrates that the biotic interactions between species provide a driving force resulting in endless evolution for some species even if the physical environment is unchanged.
is is because he neglects the Lorenz matrix (Lorenz force). If considering the Lorenz force, the population flow on a landscape is not directly down the gradient of the potential function. It also swirls (see red thin arrow in Figure 1). e above scientific understanding indicates that these three components exist in general stochastic dynamical systems, but this decomposition lacks rigorous mathematical proof. is paper is the first to prove that the deterministic part of general high-dimension stochastic dynamical systems can be decomposed into three components: the diffusion force, gradient of the potential function, and curl flux. Our previous work proved this structure for the low-dimension case (when the dimension n � 1, 2). On this basis, we prove this structure for the high-dimension case (when n ≥ 4, and (n(n − 1)/2) is an even number, i.e., n � 4, 5, 8, 9, . . . , 4i, 4i + 1, . . ., when i � 1, 2, 3, . . .). Apart from theoretical significance, our result has important guiding significance for applications in both mathematics and subjects such as biology and physics. e potential function provides intuitive and global landscapes. Real dynamical systems are complex and usually have more than one steady state, so the potential function has a wide range of applications in real dynamical systems. For example, Hu and Xu [24] studied the phenomenon of multi-stable chaotic attractors existing in generalized synchronization for a driving and response system named Rössler system. Angeli and Sontag [25] studied the emergence of multi-stability and hysteresis in those monotone input/output systems that arise, under positive feedback, starting from monotone systems with well-defined steady-state responses. Liu and You [26] studied multi-stability, existence of almost periodic solutions of a class of recurrent neural networks with bounded activation functions and all criteria they proposed can be easily extended to fit many concrete forms of neural networks such as Hopfield neural networks, or cellular neural networks, etc.. e potential function has provided a general and unified perspective for researchers to investigate different types of dynamical systems.  Figure 1: Evolution is described as populations moving on landscapes. is is represented as either (a) a maxima on a landscape of fitness; or (b) a minima on a landscape of fitness potential. ey are different ways to visualize the same process. e rest of this paper is organized as follows. Section 2 introduces Ao decomposition for general stochastic differential equations and proposes our problem: proving the equivalence of the Langevin equation and the equation after Ao decomposition. In Section 3, we reduce this problem to proving the existence of solutions for first-order partial differential equations, and we accomplish this proof in Section 4.

A-Type Decomposition for General Stochastic
Differential Equations e Langevin equation in physics, which has the form of a general stochastic differential equation, is usually a more accurate description of physical processes than the purely deterministic one [27][28][29][30]. Here, we use the physicists' notation for the noise, and we can write this equation in the form (1) We discuss this equation in n-dimensional real Euclidean space. e state variable q � (q 1 (t), q 2 (t), . . . , q n (t)) is a function of time t, and the component functions q i (i � 1, 2, . . . , n) of the state variable q � (q 1 , q 2 , . . . , q n ) are independent. We assume that f(q, t) is an infinitely differentiable smooth function. e noise ζ(q, t) is a function of t and the state variable q, and is almost nowhere differentiable. We consider the case that ζ(q, t) is n-dimensional white Gaussian noise with mean and covariance e superscript τ denotes the transpose of a matrix (vector), δ(t − t ′ ) is the Dirac delta function, 〈·〉 indicates the average over the noise distribution, and the diffusion matrix D(q) is symmetric and positive semi-definite.
Our main problem is to prove the equivalence of equations (1) and (4). We can also prove that φ(q(t)) in equation (4) is a potential function.

Reduction of Problem into Partial Differential Equations (PDEs)
To show that equation (1) is equivalent to equation (4), we first show that equation (4) implies equation (1). To this end, we assume that the function matrix [S(q) + A(q)] is invertible and the components q i (i � 1, 2, . . . , n) of the state variable q � (q 1 , q 2 , . . . , q n ) τ are independent. If they are not independent, the dimension can be reduced to n ′ , n ′ < n, until they are independent. erefore, the equations of this system are linearly independent. Equation (4) can be straightforwardly transformed to where ζ(q, t) is noise that takes the form ζ(q, t) � [S(q) + A(q)] − 1 ξ(q, t). To match equation (1), we can then set f(q) � − [S(q) + A(q)] − 1 ∇φ(q). Notice that with the explicit representation of ζ(q, t) in terms of S(q), A(q), and ξ(q, t), as well as equation (6), we can calculate Comparing the above two calculations with equations (2) and (6), we see that we have obtain S(q), A(q), and φ(q) from the general dynamic equation (1). We propose heuristic inference. While not a rigorous mathematical proof, it can lead to a reformulation of the problem into PDEs. e main idea of this heuristic inference is that equations (1) and (4) can describe the same dynamical behaviors in R n . Hence we may replace _ q in equation (4) by the right side of equation (1) to obtain Regarding t as a parameter in q(t), the above equation can be written as which has a deterministic part that is differentiable up to an arbitrary order, and a random part that is nondifferentiable everywhere. From the point of view of physics, the two kinds of noises ζ(q, t) and ξ(q, t) have the same source. Inspired by this, we may assume that we can establish a classification, is subjective decomposition is the key to understanding Ao decomposition, which results in the consistency of stable points between a stochastic dynamical system and the corresponding dynamical system. erefore, the generalized Lyapunov function of the stochastic dynamical system is equivalent to the Lyapunov function of the corresponding dynamical system. It must be noted that the A-type integral derived from Ao decomposition is a new integral that is different from the Itô and Stratonovich integrals. In one dimension, the A-type integral is simplified to the α-type [33], where α � 1 (Itô corresponds to α � 0, and Stratonovich to α � 0.5). In the high-dimensional case, the A-type integral is not usually the α-type [34].
Combining equations (3) and (6) with (13), we obtain which implies From the physical point of view, equation (16) is a generalized Einstein relation in greater than one dimension. From equation (16), we have where the symmetric part of . is is the diffusion matrix D(q) defined in equation (3). Hence we can rewrite the identity as where Q(q) is an anti-symmetric unknown matrix function and I is the identity matrix. Substituting equation (19) in equation (15), we obtain From equation (15), it is easy to obtain that if _ q � f(q * ) � 0, φ(q * ) � 0. Moreover, by equation (15) we have 4 Journal of Mathematics us φ(q) satisfies _ φ(q) ≤ 0 for all q ∈ R n , and it is proven that φ(q) is a potential function.
Assuming equation (1) holds true, equation (3) is given, and thus D(q) is known. We see that to obtain equation (4), we just have to show that there exists an anti-symmetric matrix Q(q) and potential function φ(q) that satisfy equation (20). Assuming basic integrability conditions on f(q), D(q), and Q(q), by the classical Helmholtz decomposition we obtain that it suffices to show that the curl part of the vector field where equation (22) is a family of (n(n − 1)/2) first-order quasilinear partial differential equations for the coefficients of Q(q) in equation (20). We notice that according to the above heuristic inference, equation (22) is a sufficient condition for equations (1) ⇒ (4). In fact, if equation (22) holds true, then by Helmholtz decomposition, there exists a function φ � φ(q) such that equation (20) holds true, with the anti-symmetric matrix Q(q) from (22). Moreover, with D(q) from equation (3) and Q(q) from equation (22), we can construct the matrix is anti-symmetric, and S(q) and A(q) satisfy equations (15) and (16). us we can construct the noise ξ(q, t) from equation (13), which, together with equation (12), implies that we can construct equation (4) from equation (1).
Our problem has now been reduced to proving the existence of solution Q(q) to equation (22), which is a first-order PDE. e rest of the paper is dedicated to the investigation of this first-order PDE system in dimension n ≥ 4.

Existence of Solutions to First-Order
Quasilinear Partial Differential Equations in Dimension n ≥ 4 Obviously, q � (q 1 , q 2 , . . . , q n ), n ≥ 4. Assume that From equation (3), we can assume that By equation (22), we can assume erefore, according to the matrix-valued cross-product rule, equation (22) can be transformed to According to the composite function derivation rule, we obtain Journal of Mathematics where superscripts (1), (2), . . . , (n) denote the partial derivatives corresponding to q 1 , q 2 , . . . , q n , respectively. is is obviously a first-order n-dimensional quasilinear system of partial differential equations consisting of (n(n − 1)/2) equations and (n(n − 1)/2) unknown functions. Our goal is to prove the existence of solutions for PDEs (27). We first consider the matrix form of PDEs (27), e independent variables are x � (q 1 , . . . , q i ) and y � (q i , . . . , q n ), where 1 < i < n. If A has no real eigenvalues at any point in a region, PDEs (28) are elliptic in this region, and they obey the rule that the equations do not explicitly contain time. Because A is a real matrix, this may occur only when (n(n − 1)/2) is an even number. Obviously, when the multiplicities of eigenvalues of matrix A at each point (x, y) are constant in the whole region, then the order of every subblock of the Jordan standard form of A is constant in the whole region, such that there exists a nonsingular matrix T(x, y) satisfying where en we assume the following: (i) T and A belong to function space C 1 α (G); (ii) e order of every Jordan sub-block of matrix J � TAT − 1 is constant in the whole region G; (iii) e eigenvalue λ j (x, y) ∈ C α (G).
Let v � TR. Equation (28) can be transformed to v x + Jv y + zero − order term � 0. (31) en we study the solution of system is system can be decomposed into a sub-system with the form where j � 0, 1, . . . , s, and v j 1 , v j 2 are real vector functions whose dimensions equal the order of matrix J i . Let where i is the imaginary unit. en equation (33) can be written in the form Using operator notation, When j � 0, the sub-system can be decomposed into p equations like where the value of λ is selected from λ 1 , . . . , λ p . is case has been solved [35]. When j ≠ 0, the sub-system can be transformed to where erefore, we consider the system where e has the form of equation (38). We assume that the pair of eigenvalues (λ, λ) is r-fold and the corresponding linear independent eigenvector is only one, so the complex vector ω in PDEs (39) is r-dimensional. Obviously, without loss of generality, we can assume λ in PDEs (39) satisfies Imλ > 0, and PDEs (39) are uniformly elliptic, We divide both sides of PDEs (39) by 1 − iλ to obtain where I is a unit matrix of order r. Because the linear transformation ((1 + iξ)/(1 − iξ)) maps the half-plane Imξ > ε 0 to disk |ξ| ≤ ρ < 1, we obtain the function 6 Journal of Mathematics which satisfies |q 0 (z)| ≤ ρ < 1, where ρ is some positive constant.
A quasi-diagonal matrix is a lower triangular matrix, we set (i) a 0 represents an element of the main diagonal; (ii) a j , 1 ≤ j ≤ n − 1, represent elements of the jth diagonal under the main diagonal.
Because the coefficient matrix of ω z and ω z in PDEs (41) is quasi-diagonal, PDEs (41) can be written as where Q(z) is quasi-diagonal, and the element q 0 in main diagonal of Q(z). e first equation of PDEs (41) is which is a Beltrami equation. Because λ ∈ C α (G), q 0 ∈ C α (G), we can extend q 0 such that it belongs to C α (θ) and maintain 0 outside a large enough circle. With |q 0 | ≤ ρ < 1, we obtain the solution ξ(z) ∈ C 1 α (θ) of Beltrami equation (45) [36].
Under the coordinate transformation ξ � ξ(z), PDEs (44) change to standard form, If we let Q represent the coefficient matrix of ω ξ , and still use z as the independent variable, we have A. Douglis derived the quasi-diagonal form of Q by introducing the algebra of hypercomplex numbers [36,37].
Definition 2 (see [36]). a � r− 1 k�0 a k e k is called a hypercomplex number, where e is defined by equation (38), a k is a complex number, and a 0 is the complex number part of a, and r− 1 k�1 a k e k is the nilpotent part of a. Note that |a| � r− 1 k�0 |a k |, where a k is the kth component of a. A hypercomplex function is a map from the plane into this algebra, and it has the form where each ω k is complex-valued.
Using Definition 2, we can write PDEs (47) as Note the differential operator, where q m is the mth column vector of q. Using the nilpotency of e, we have erefore, PDEs (49) can be written as We define a generating solution.

Definition 3.
A hypercomplex function space that has bounded continuous derivatives up to order k defined in set N is represented by B k (N), and a hypercomplex function space whose k-order derivative with index α is € Holder A hypercomplex function t(z) is called a generating solution of operator D if: We prove the existence of a generating solution to PDEs (32). We assume that and q k (z) can be extended to θ such that they all belong to C 0 α (θ) and are equal to zero outside a large enough circle. Let

Journal of Mathematics
where J G is an integral operator, By the property of operator J G and the assumption on q k (z), we obtain By equation (52), we obtain erefore, t(z) is the generating solution. t(z) has the property of a positive constant M such that where (1/(t(ζ) − t(z))) is another sign of (t(ζ) − t(z)) − 1 . e inverse exists because the complex part ζ − z of t(ζ) − t(z) is not zero.
Next, we consider two corresponding boundary value problems of original nonhomogeneous PDEs (28), the nonlinear Riemann boundary value problem and nonlinear Riemann-Hilbert boundary value problem. PDEs (26) can be written in the following form in the sense of Douglis algebra: where differential operator where q(z) is a known nilpotent hypercomplex function, and where F k is a known complex value function of all its variables. Assume that I(z, ω) � I(z, z, ω, ω) is a hypercomplex function of independent variable z and hypercomplex variable ω. Define the Gateaux first-order differential of I about ω, ω, We can similarly define the second-order Gateaux differential (δ h ω ) 2 I, δ h ω δ h ω I, (δ h ω ) 2 I, and so on. We utilize Γ to represent a simple smooth closed contour in the complex plane Γ, whose positive direction is counterclockwise. It divides θ into bounded interior region G + and external unbounded region G − . Assume that M τ, ω (1) , is a known hypercomplex function on variable τ ∈ Γ, with hypercomplex elements Assume that g(τ) is a hypercomplex function that satisfies the € Holder condition on Γ, whose complex part g 0 (τ) is not zero forever on Γ. We also introduce the integer notation Now we can introduce the corresponding nonlinear Riemann boundary value problem.

Definition 4 (Nonlinear Riemann boundary value problem).
Assume that G + is a bounded and simply connected region in plane θ, whose boundary Γ is a smooth closed curve, and the positive direction of Γ causes G + to be located to the left. Note the complement of G + + Γ as G − , where the origin of coordinates is located in G + . In the whole plane θ, we seek the normal block solution ω(z) to PDEs (61), such that a nonlinear boundary value in Γ that satisfies has definite order m − n at infinity, where m is an integer. en we consider the linear Riemann boundary value problem, Assume that A(z), B(z), C(z) ∈ L p,2 (θ), p > 2, are known hypercomplex functions. Note 8 Journal of Mathematics and known hypercomplex function c(τ) ∈ B 0 α (Γ) on Γ. en we represent the generating solution of differential operator D by t(z). Before stating our main theorem and proof, we introduce four lemmas. e proofs of these lemmas can refer to Appendix.

Lemma 1.
e Cauchy-type integral is a block hyperanalytic function that is equal to 0 at infinity, and the estimate establishes where N (1) is a positive constant related to α, q(z), and Γ. In addition, the boundary value condition satisfies on Γ,

Lemma 2. Assume hypercomplex functions A(z), B(z), f(z) ∈ L p,2 (θ), p > 2. e integral operator K is defined by
where f is a hypercomplex function in the whole plane θ. Hypercomplex functional (Kf)(z) satisfies: (iv) Hypercomplex functional ω(z) � (Kf)(z) satisfies the following system in the Sobolev sense: In (i)-(iii), M is a positive constant only relative to q, A, B, p, and R. e positive number in (ii) and (iii) is According to Lemma 2, operator (Kf)(z) is zero at infinity and continuous in the whole plane θ. en we can establish the expression and estimate of the solution of boundary value problem (69).

Lemma 3. Boundary value problem (69) has a unique solution, ω(z) � φ(z) + K(− Aφ
where and K is an integral operator defined by Lemma 2. Next, we introduce two estimates of solution ω(z).

Lemma 4. For solution ω(z) of boundary value problem
(69), the following estimates hold true: where N is a positive constant only relative to p, q(z), A(z), B(z), and Γ.
e same as above, if hypercomplex function For solution ω(z) of (69), from estimates (81), we can deduce that |ω, θ| 0,α ≤ 2N |c, Γ| 0,α +|C, θ| p,2 . (83) Now, we go back to the research on seeking solutions for nonlinear Riemann boundary value problems (61) and (68). e corresponding hypercomplex function is represented by X(z), which means the determined hyperanalytic function defined in the whole plane θ, and it satisfies boundary value condition X + (τ) � g(τ)X − (τ) on Γ, and has − n order at infinity. Because it has complex number parts that are not zero everywhere, it has inverse (1/X(z)).
If m ≤ − 1, when |z| ⟶ ∞, ω(z) ⟶ 0. If m ≥ 0, there must exist a hyperpolynomial whose order is within m, where all a j are hypercomplex constants, such that under the transformation A new hypercomplex function ω(z) satisfies a similar system and boundary value condition, and when |z| ⟶ ∞, ω(z) ⟶ 0.
at only needs adding some degenerating condition on hypercomplex function F(z, ω) and its first order and second order Gateaux differential at infinity. We omit the specific condition.

Theorem 1. Under the above assumptions, if positive constant Q in inequality (91) satisfies
where N is the positive constant appearing on the right side of estimate equations (81) or (83), the solution of nonlinear boundary value problem (89), which can be written as ω(z) ∈ B 0 α (G + + Γ) ∩ B 0 α (G − + Γ), must exist, and it can be constructed by a successive approximation and continuity method.
Before the proof, we establish an estimate equation.  Proof. We consider a family of nonlinear Riemann-Hilbert boundary value problems with real parameter λ, 0 ≤ λ ≤ 1: We want to prove that for every λ: 0 ≤ λ ≤ 1, the solution ω(z; λ) of boundary value problem (118) exists, and belongs to B 1 ] (G + + Γ); then ω(z; 1) is the solution of the above Riemann-Hilbert boundary value problems (61) and (111).

Conclusion
We revealed a fundamental structure of general stochastic dynamical systems proposed by Ao et al.. We demonstrated a scientific understanding of three essential components: the potential function, friction matrix S(q), and Lorenz matrix A(q). Our goal was to prove the equivalence between general stochastic differential equations and equations after A-type decomposition, and then we could assert that the above elements are fundamental components of general stochastic dynamical systems.
is problem can be transformed to proof of the existence of solutions for first-order quasilinear partial differential equations. en we mathematically proved the existence of solutions for these equations. Specifically, when dimension n satisfies n ≥ 4, and (n(n − 1)/2) is an even number (n � 4, 5, 8, 9, . . . , 4i, 4i + 1, . . ., when i � 1, 2, 3, . . .), the existence of the generated solutions of the homogeneous equations corresponding to these equations was proved by introducing the hypercomplex algebra proposed by Douglis. en, by successive approximation and continuous methods, we proved the existence of solutions of the Riemann boundary value problem and Riemann-Hilbert boundary value problem corresponding to first-order quasilinear partial differential equations. erefore, we proved this fundamental structure of general stochastic dynamical systems for the high-dimensional case.

Appendix
We begin with some definitions and properties of hypercomplex functions, to enable readers to better understand hypercomplex calculation. As mentioned before, hypercomplex numbers and functions are defined in Definition 2, and the hypercomplex function space and corresponding generated solutions are defined in Definition 3.
We briefly discuss norms of hypercomplex numbers in our algebra. For a, as given by Definition 2, Douglis [36] defined the norm Furthermore, writing a as a � a 0 + E, where E is the nilpotent part of a, the inverse of a is where a 0 ≠ 0. erefore, we also have the inequality Second, we list some inequalities concerning the generated solution. We denote generic constants by M(·). We have t x (z) , t y (z) ≤ M(e, f), (A.6) In each case above, M(e, f) is a constant depending on the bounds on f and the derivatives of t.
ird, we state some properties of hypercomplex functions. Theorem A.2 (Plemelj-Privalov theorem [41]). If Γ is a circle, then the singular integral operator (also called a Hilbert transform) given by the Cauchy principal value integral, behaves invariantly with respect to the class of € Holder continuous functions, denoted by C 0,α (Γ) for 0 < α < 1.
Furthermore, Sokhotski proposed the Sokhotski-Plemelj formula [42], and Plemelj proved it [41]. Let t 0 � ξ 0 + iη 0 represent an arbitrary fixed point on L excluding the endpoint when f(t 0 ) ≠ 0. If f(t) satisfies the following condition on neighborhood ε > 0 of every point on L: for two arbitrary point t 1 and t 2 on these neighborhoods, there exists a constant A satisfies f t 1 − f t 2 ≤ A t 1 − t 2 λ , 0 < λ ≤ 1, (A.10) limit of f(z) exists. When z approaches point t 0 from the left side of L, this limit is named F + (t 0 ), and conversely, when z approaches point t 0 from the right side of L, this limit is named F − (t 0 ). ey satisfy where the symbol ′over the integral denotes the Cauchy principal value integral. en we give proofs of Lemmas 1-4.

Proof of Lemma 1
Proof. By the Sokhotski-Plemelj formula (see eorem A.3), the Cauchy-type integral φ(z) is a block hyperanalytic function and satisfies boundary value condition (74). Next, we discuss the property by which boundary value is is a natural generalization of the Plemelj-Privalov theorem ( eorem A.2 in Appendix) for a usual analytic function. Finally, we start to prove estimate equation (73). Assume that τ 0 is an arbitrary point on Γ, z ∈ G + , and consider hypercomplex function On any single-valued branch in G + , by the Plemelj-Privalov theorem and the properties of t(z), it is easy to know that the boundary value of this function, satisfies the € Holder condition on Γ. Similar to an analytic function, we can prove in the whole region G + , dt(ζ). where N (5) is a positive constant, which depends linearly on the € Holder coefficients of the hypercomplex function c(τ). For point z in region G − , we can similarly prove the above inequalities. erefore, we obtain estimate equation (73).