A higher dimensional fractional Borel-Pompeiu formula and a related hypercomplex fractional operator calculus∗

In this paper we develop a fractional integro-differential operator calculus for Clifford-algebra valued functions. To do that we introduce fractional analogues of the Teodorescu and Cauchy-Bitsadze operators and we investigate some of their mapping properties. As a main result we prove a fractional Borel-Pompeiu formula based on a fractional Stokes formula. This tool in hand allows us to present a Hodge-type decomposition for the fractional Dirac operator. Our results exhibit an amazing duality relation between left and right operators and between Caputo and Riemann-Liouville fractional derivatives. We round off this paper by presenting a direct application to the resolution of boundary value problems related to Laplace operators of fractional order.


Introduction
Clifford analysis offers a higher dimensional generalization of the classical theory of complex holomorphic functions. Its tools can be applied to several different areas, for instance to quantum mechanics, quantum field theory [15], projective geometry, computer graphics [30], neural network theory [3] and to many other areas of physics and engineering [17]. The corresponding analogy of the class of complex holomorphic functions is that of monogenic functions. These are the null solutions to the Dirac operator. The latter operator factorizes the Laplace operator and provides a first order generalization of the well-known Cauchy-Riemann operator in complex analysis (see [5,8]).
A main tool that Clifford holomorphic function theory uses in the treatment of boundary value problems is the Teodorescu operator, which is the right inverse of the Dirac operator. Properties and applications of the hypercomplex Teodorescu operator have been studied by many authors (see for instance [29] for a list of references). In the context of quaternionic and Clifford analysis, K. Gürlebeck and W. Sprößig studied among many others particular mapping and regularity properties of this integral operator. Furthermore, they studied its connections to elliptic boundary value problems (see [17]). Additionally, in [4] the authors also investigated some interesting connections between the Teodorescu operator and Hermitian regular functions. An extension to the time-dependent case addressing the heat and the Schrödinger operator has been presented subsequentially in [6].
Another central aspect that appears in the classical vector calculus and in generalized Clifford holomorphic function theories is the Helmholtz decomposition of L 2 -spaces. Actually, in classical three-dimensional vector analysis it is nothing else than the decomposition of an arbitrary sufficiently regular vector field into the sum of a divergence free field (having a vector potential) and a curl free vector field (having a scalar potential). This particular space decomposition together with the Teodorescu operator calculus provides a very elegant resolution toolkit for boundary value problems in the corresponding scales of Hilbert-Sobolev spaces. For more details we also refer to the survey paper [27]. For the time-dependent case, see for instance [7,22,23].
A parallel development over the last years consists of a rapidly increasing interest in the theory of derivatives and integrals of non-integer order. Apart from several applications of fractional order models, as for example, to kinetic theories, statistical mechanics, to the dynamics in complex media, and to many other fields (see [28] and the references indicated therein), those methods provide an important counterpart and extension of the classical integer order models. The advantage of fractional models consists in the possibility of using fractional derivatives to describe the memory and hereditary properties of various materials and processes. Another field of application consists in addressing differential equations related to flows with permeable boundaries, such as for instance dam-fill problems which provides a further important motivation to develop three-dimensional generalizations of harmonic and Clifford analysis tools for the fractional setting. Preceding work pointing in this direction can be found in [18,19] where the fractional p-Laplace equation has been treated.
Behind this background, the development of links between Clifford analysis and fractional differential calculus represents a very recent topic of research. In particular, some first steps in the direction of an introduction of a fractional Clifford analytic function theory have been made in [10][11][12]14]. In these papers, the authors determined series representations for the fundamental solution related to some stationary and non-stationary fractional Dirac-type operators. The knowledge of explicit representation formulas of these fundamental solutions represents a corner stone in the development of a fractional version of Clifford analysis. The latter functions serve as kernels for fractional integral operators, such as the fractional Teodorescu operator that we are going to introduce and to investigate in this paper.
The aim of this paper is to apply the fundamental solutions obtained in [10,12] in order to develop the fundamentals of a fractional operator calculus related to the fractional Dirac operator that depends on a vector of fractional parameters α = (α 1 , . . . , α n ) with α i ∈ ]0, 1], i = 1, . . . , n. We introduce fractional analogues of the Teodorescu operator and of the Cauchy-Bitsadze operator, and we investigate some important mapping properties. Moreover, we present a Hodge-type decomposition for the fractional Dirac operator defined via left Caputo fractional derivatives. The results that we obtain exhibit an amazingly interesting "double duality" between left and right operators and between Caputo and Riemann-Liouville fractional derivatives. This double duality appears in a non-trivial generalization of the Stokes formula as well as in the fractional Borel-Pompeiu formula and in the Hodge-type decomposition that we are going to present subsequentially. Throughout the paper we show that we can always re-obtain the results of the classical function theory for the Dirac operator when switching to the limit case when α = (1, . . . , 1). The analogous of the results presented in this paper for the case of the time-fractional parabolic Dirac operator can be found in [13].
The structure of the paper reads as follows. In the Preliminaries section we recall some basic definitions from the fractional calculus, special functions, and Clifford analysis. In Section 3, we present the fundamental solutions of the fractional Laplace and Dirac operators in R n , defined by left Riemann-Liouville and Caputo fractional derivatives. Moreover, we prove that these functions belong to the function space L 1 (Ω) under certain conditions. Throughout the whole paper we assume that Ω is a bounded open rectangular domain. In Section 4, we introduce and study the main properties of the fractional analogues of the Teodorescu operator and of the Cauchy-Bitsadze operator. Finally, in Section 5 we present a Hodge-type decomposition for the L q -space, where one of the components is the kernel of the fractional Dirac operator defined in terms of left Caputo fractional derivatives. This decomposition represents a main result in the paper apart from proving the generalizations of the Borel-Pompeiu formulae in the context of Caputo derivatives. In the analysis of the mapping properties and the regularity properties there still appear some further peculiarities that require special attention. We round off this paper by giving an immediate application to the resolution of boundary value problems involving the fractional Laplace operators.

Fractional calculus and special functions
Let a, b ∈ R with a < b let α > 0. The left and right Riemann-Liouville fractional integrals I α a + and I α b − of order α are given by (see [21]) By RL D α a + and RL D α b − we denote the left and right Riemann-Liouville fractional derivatives of order α > 0 on [a, b] ⊂ R, which are defined by (see [21]) Here, m = [α] + 1 and [α] means the integer part of α. Let C D α a + and C D α b − denote, respectively, the left and right Caputo fractional derivative of order α > 0 on [a, b] ⊂ R, which are defined by (see [21]) We denote by I α a + (L 1 ) the class of functions f that are represented by the fractional integral (1) of a summable function, that is f = I α a + ϕ, with ϕ ∈ L 1 (a, b). A description of this class of functions is given in [26].  [25,26]). Removing the last condition in Theorem 2.1 we obtain the class of functions that admit a summable fractional derivative.
= 0 which implies that the first term in the series expansion of (x − a) ν−1 E µ,ν (k(x − a) µ ) vanishes. Therefore, the derivation rule (12) must be replaced in these cases by the following derivation rule: Remark 2.4 For ν = p with p = 1, . . . , m, we have that C D α a + ((x − a) p−1 ) = 0 which implies that the first term in the series expansion of (x − a) ν−1 E µ,ν (k(x − a) µ ) vanishes. Therefore, the derivation rule (13) must be replaced in these cases by the following derivation rule: The approach presented in this paper is based on the Laplace transform and leads to the solution of a linear Abel integral equation of the second kind.
has a unique solution Now we recall the formula for fractional integration by parts for 0 < α < 1 and x ∈ [a, b] (see [ We end this section by recalling an important result about the boundedness of the fractional integrals I α a + and I α b − (see Theorem 3.5 in [26]). Theorem 2.6 If 0 < α < 1 and 1 < p < 1 α then the operators I α a + and

Clifford analysis
Let {e 1 , · · · , e n } be the standard basis of the Euclidean vector space in R n . The associated Clifford algebra R 0,n is the free algebra generated by R n modulo x 2 = −||x|| 2 e 0 , where x ∈ R n and e 0 is the neutral element with respect to the multiplication operation in the Clifford algebra R 0,n . The defining relation induces the multiplication rules where δ ij denotes the Kronecker's delta. In particular, e 2 i = −1 for all i = 1, . . . , n. The standard basis vectors thus operate as imaginary units. A vector space basis for R 0,n is given by the set {e A : A ⊆ {1, . . . , n}} with e A = e l1 e l2 . . . e lr , where 1 ≤ l 1 < . . . < l r ≤ n, 0 ≤ r ≤ n, e ∅ := e 0 := 1. Each a ∈ R 0,n can be written in the form a = A a A e A , with a A ∈ R. The conjugation in the Clifford algebra R 0,n is defined by a = A a A e A , where e A = e lr e lr−1 . . . e l1 , and e j = −e j for j = 1, . . . , n, e 0 = e 0 = 1. Each non-zero vector a ∈ R n has a multiplicative inverse given by a ||a|| 2 . An R 0,n −valued function f over Ω ⊆ R n has the representation f = A e A f A with components f A : Ω → R 0,n . Properties such as continuity or differentiability have to be understood componentwise. Next, we recall the Euclidean Dirac operator D = n j=1 e j ∂ xj . This operator satisfies For more details about Clifford algebras and basic concepts of its associated function theory we refer the interested reader for example to [8].

Fundamental solutions revisited
In [10] and [12] the authors considered the so-called three-parameter fractional Laplace and Dirac operators defined in terms of the left Riemann-Liouville and Caputo fractional derivatives, and obtained families of eigenfunctions and fundamental solutions for both operators. In this section we present the generalization of these results for R n . Let Ω = n i=1 ]a i , b i [ be any bounded open rectangular domain, let α = (α 1 , . . . , α n ), with α i ∈ ]0, 1], i = 1, . . . , n, and let us consider the n-parameter fractional Laplace operators RL ∆ α a + and C ∆ α a + defined over Ω by means of the left Riemann-Liouville and left Caputo fractional derivatives, respectively, given by Associated to them there are the corresponding fractional Dirac operators RL D α a + and C D α a + defined by For i = 1, . . . , n the partial derivatives RL are the left Riemann-Liouville and Caputo fractional derivatives (3) and (5) of orders 1 + α i and 1+αi 2 , with respect to the variable [10]), and C ∆ α [12]). Due to the nature of the eigenfunctions and the fundamental solution of these operators we additionally need to consider the variable x = (x 2 , . . . , x n ) ∈ Ω = n i=2 ]a i , b i [, and the fractional Laplace and Dirac operators acting on x defined by We start by addressing the Caputo case. Consider the eigenfunction problem where λ ∈ C, and where we suppose that v(x) = v(x 1 , . . . , x n ) admits a summable fractional derivative v (x) in the variable x 1 , and belongs to I 1+αi to both sides of the previous equation and taking into account Now, applying successively the fractional integrals I 1+αj a + j , with j = 2, . . . , n, to both sides of the previous equation, applying Fubini's theorem, and rearranging the terms, we get where g 0 and g 1 are the Cauchy initial conditions given by We observe that the fractional integrals in (22) are Laplace-transformable functions. Therefore, we may apply the (n − 1)-dimensional Laplace transform with respect to x 2 , . . . , x n : Taking into account its convolution and operational properties (see [9,21]), we obtain the following relations for each term in (22): Combining all the resulting terms and multiplying by n p=2 s 1+αp p we obtain the following second kind homogeneous integral equation of Volterra type: where G(x 1 , s) = G 0 ( s) + (x 1 − a 1 ) G 1 ( s) and G k ( s) = L {g k } (s) with k = 0, 1. Using (16), we have that the unique solution of (29) in the class of summable functions is: which involves the two-parameter Mittag-Leffler function. Due the convergence of the integrals and the series that appear in (30), we can interchange them and rewrite (30) in the following way: In order to cancel the Laplace transform, we need to take into account its distributional form in Zemanian's space (for more details about generalized integral transforms see [31]) and the following relation: where k = 0, 1. Therefore, applying the multinomial theorem and after straightforward calculations we get the following solution of (21): From the previous calculations we obtain the following results in R n , which generalize Theorem 3.1 and Theorem 4.1 in [12].
Theorem 3.1 A family of eigenfunctions of the fractional Laplace operator C ∆ α a + is given by We give a direct proof of the theorem in order to confirm that (34) is indeed the solution of (21 + C ∆ α a + to (34) and using the series expansion of the Mittag-Leffler function (10), we get Rearranging the terms of the series we obtain Corollary 3.2 A family of fundamental solutions for the fractional Laplace operator C ∆ α a + is obtained by considering λ = 0 in (34): For the fractional Dirac operator C D α a + we can obtain a family of fundamental solutions by applying the operator C D α a + to the family of fundamental solutions of the operator C ∆ α a + . Theorem 3.3 A family of fundamental solutions of the fractional Dirac operator C D α a + (acting on the left or on the right) is given by and for i = 2, . . . , n Now we present the corresponding results for the Riemann-Liouville case. First we obtain the eigenfunctions associated to the operator RL ∆ α a + satisfying RL ∆ α a + v(x) = λv(x), where λ ∈ C, and v(x) = v(x 1 , . . . , x n ) admits a summable fractional derivative RL Theorem 3.4 A family of eigenfunctions of the fractional Laplace operator RL ∆ α a + is given by x).
The proof of Theorem 3.4 is similar to the proof of Theorem 3.1, however, it takes into account the composition rule (7). For the case n = 3, see [10].
Corollary 3.5 A family of fundamental solutions for the fractional Laplace operator RL ∆ α a + is obtained by considering λ = 0 in (39): For the fractional Dirac operator RL D α a + we can obtain a family of fundamental solutions by applying the operator RL D α a + to the family of fundamental solutions of the operator RL ∆ α a + .
Theorem 3.6 A family of fundamental solutions of the fractional Dirac operator RL D α a + (acting on the left or on the right) is given by and for i = 2, . . . , n Remark 3.7 From (40) or (35) it is possible to obtain the fundamental solution of the Euclidean Laplace operator in R n when α = (1, . . . , 1). Let us consider only the Riemann-Liouville case (the Caputo case can be treated similarly). We know that the fundamental solution of the Euclidean Laplace operator in R n , n ≥ 3, is given (up to a constant) by ||x − a|| −(n−2) . The case n = 2 can also be treated but we restrict ourselves to only present the case n ≥ 3 in detail. First we need to obtain the power series expansion of the fundamental solution of the Laplace operator in R n . Considering the binomial series x − a 2 < 1. Now, putting α = (1, . . . , 1) and f 1 the null function in (40) we obtain From a comparison of (44) and (45) we observe that we have to find a function f 0 ( x) such that We observe that the function f 0 ( x) = x − a −(n−2) satisfies (46). First we recall that the p-th powers of the m-dimensional Euclidean Laplace satisfy (see [2, (1.5)]) with r = x , x ∈ R m , p ∈ N and k ∈ Z. Therefore, for m = n − 1 and k = 2 − n we obtain Comparing (46) and (47) we conclude that we have to show that n−2 This equality is true and can be proved by using well-know relations for the Gamma function and taking into account that .
Therefore, we conclude that the equality (46) is satisfied when f 0 ( x) = x − a −(n−2) . As expected, on the basis of considering this same function, together with f 1 being the null function, we may obtain from (41) and (36) the fundamental solution for the Euclidean Dirac operator in R n , n ≥ 3, when α = (1, . . . , 1).
In the following section we introduce fractional versions of the Teodorescu and Cauchy-Bitsadze operators where the kernel of these operators is the fundamental solution C G α + . Before we proceed to the development of the operator calculus we present the following auxiliar results.
Theorem 3.8 For functions g 0 and g 1 such that the fundamental solution C G α + belongs to L 1 (Ω).
Proof: From (35) we have Relying on the series expansion of the Mittag-Leffler function (10), and the fact that the series and integrals that are involved are absolutely convergent, we derive that where Ω = n j=2 ]a j , b j [, and where g 0 , g 1 , are chosen such that the integrals over Ω are finite for each i = 1, . . . , n. Let us denote by C 0 and C 1 the corresponding maximum values over i. Moreover, computing the integrals with respect to x 1 leads to the following inequality Moreover, since 0 < α 1 ≤ 1, we get the final estimate The last expression is a finite quantity in view of a 1 < b 1 .
In a very similar way we can prove the following result for the fundamental solution of C D α a + .
Theorem 3.9 For functions g 0 and g 1 such that the fundamental solution C G α + belongs to L 1 (Ω).

Fractional Teodorescu and Cauchy-Bitsadze operators
In this section we introduce and study the main properties of the fractional analogues of the classical Teodorescu and Cauchy-Bitsadze operators described in [17]. We start by proving fractional analogues of the Stokes formula and the Borel-Pompeiu formula in a rectangular open rectangular domain of the form Ω = n i=1 ]a i , b i [. From now on C D α b − denotes the right Caputo fractional Dirac operator, which is given by where, for i = 1, . . . , n, the partial derivative where dσ(x) = n(x) dΩ, with n(x) being the outward pointing unit normal vector at x ∈ ∂Ω, where dΩ is the classical surface element, and dx represents the n-dimensional volume element.
Before we give a proof of theorem we observe that in (50) the operator C D α b − acts on the right and the operator RL D α a + acts on the left, which is specific of the Clifford analysis setting because of the lack of commutativity. Proof: Suppose that f and g satisfy the above mentioned conditions. From (49) and (6) we obtain that Concerning the integral appearing in (51) we have where . . , x n ) and where (∂ xi f A ) (x * i , w) means that after differentiation, the variable x i is replaced by w, while the remaining variables remain unchanged. Changing the order of integration in the two inner integrals and relying on (1), we obtain that the right-hand side of (52) equals to Hence, inserting (53) into (51) we conclude that Applying now the classical Stokes formula (see [8]) to the right-hand side of (54) and applying (3), we get Therefore, from (49) we obtain the following fractional Stokes formula We notice that the Stokes's formula in the classical Clifford analysis setting has the form where D is the Euclidean Dirac operator. However, in the fractional Clifford analysis setting we obtain a more complicatedly kind of "double duality" relation. On the one hand the formula involves both the Caputo and Riemann-Liouville derivatives, and on the other hand it also involves left and right derivatives. It is also possible to obtain other versions of the fractional Stokes's formula. For example, if we consider in (51) the operator RL D α a + then we obtain the following alternative version of the Stokes's formula: .
Before we deduce our fractional Borel-Pompeiu formula, we need to understand the behaviour of the fractional Dirac operator C D α b − when the argument of the function f in (50) is translated and reflected. Denoting the translation operator by T θ f (y) := f (θ + y) and the reflection operator by R y f (y) := f (−y), and applying the definitions of the right and left Caputo fractional derivatives presented in (6) and (5) we can deduce the following relation (where the derivative is with respect to the variable y): Replacing f by C G α + (x + a − y) in (50) and integrating with respect to the variable y, we obtain the following fractional Borel-Pompeiu formula and fractional Cauchy's integral formula.
Corollary 4.2 Let g ∈ AC 1 (Ω) ∩ AC(Ω). Then the following fractional Borel-Pompeiu formula holds Moreover, if g ∈ ker RL D α a + , then we obtain the fractional Cauchy's integral formula Proof: Note that C G α + (y) is the fundamental solution of C D α a + defined only for y i > a i , i = 1, . . . , n and satisfies . . , n, due to translations and reflections, the Applying (56) with θ = x + a, leads to Ω δ(x − y) g(y) dy + The previous expression leads to the fractional Borel-Pompeiu formula Additionally, if g ∈ ker RL D α a + then the first integral of the right-hand side of the preceding expression is equal to zero. Therefore, we arrive at the fractional version of Cauchy's integral formula stated in (58).
The previous definition allows us to rewrite (57) in the alternative form Now we study some properties of the fractional integral operators C T α and C F α . We point out that in all the forthcoming results the parameter p referring to the L p -space belongs to the interval 1, 2 1−α * , with α * = min 1≤i≤n {α i }. This specific range of p results from the application of Theorem 2.6 to the fractional integrals of order 1−αi 2 , with i = 1, . . . , n, that appear in the definition of the fractional differential operators. Moreover, the parameter q of the L q -space must be chosen such that q = 2p 2−(1−α * )p , according to Theorem 2.6. If α = (1, . . . , 1), then we conclude that p ∈]1, ∞[ and q = p, as it occurs in the classical setting (see [17,Ch.3]). Before we deduce two properties of the fractional integral operators (59) and (60), we need to understand the behaviour of our fractional derivatives when the argument of the function over which we apply the derivatives is only translated. Denoting the translation by T θ f (x) := f (x + θ), and using the definition of the left Caputo fractional derivative presented in (5), we can deduce the following relation (where the derivative is with respect to the variable x): Theorem 4.6 The fractional operator C T α is the right inverse of C D α a + , i.e., for g ∈ L p (Ω), with p ∈ 1, 2 .
Proof: Note that C G α + (x) is the fundamental solution of C D α a + defined only for x i > a i , i = 1, . . . , n and satisfies C D α a + C G α + (x) = δ(x − a). With respect to the variable x, the function C G α + (x + a − y) is defined only for x i > y i , i = 1, . . . , n, therefore, the operator C D α a + is replaced by C D α y + . Taking into account the definition of C T α given in (59) and the relation (61) with θ = a − y, we obtain .
Proof: By the same reasonings used Theorem 4.6, we have from (60) that Note that the validity of the last equality is due to the fact that x ∈ Ω and y ∈ ∂Ω, i.e., the difference x − y is always non-zero.
Now we present some mapping properties of the fractional operators C T α and C F α .
Proof: Under the previous conditions, and in view of the Young's inequality for convolutions (see Theorem 1.4 in [26]) and Theorem 3.9, we obtain which leads to our result.
Now we want to study the derivatives of C T α . Before we do that we present an auxiliar result where we calculate the partial fractional derivatives of C G α + .
Theorem 4.10 The partial fractional derivatives of the fundamental solution C D α a + are given by and for k = 2, . . . , n Proof: Let us start with the proof of (62 x k only acts on the functions g 0 and g 1 .
Let us now study the derivatives of C T α .
Theorem 4.11 Let g ∈ L q (Ω), with q = 2p 2−(1−α * )p , p ∈ 1, 2 1−α * , and α * = min 1≤i≤n {α i }. The fractional partial derivatives of C T α with respect to x k satisfy the mapping property Proof: Since for k = 1, . . . , n we have then to study the derivatives of the operator C T α it suffices to study the convolution terms (64) (see [24]). The expression for the kernel of this convolution corresponds to the expressions (62) and (63). These kernels can be proved to be L 1 -functions in a similar way as it was done in the proof of Theorem 3.8, with g 0 and g 1 like in Theorem 3.8. This fact, combined with Young's inequality for convolutions (see Theorem 1.4 in [26]), leads to which in turn implies our result.
Proof: For a function f ∈ W α− 1 p ,p a + (∂Ω) we can find a function g ∈ W α,q a + (Ω) such that g = C F α f . Next, by the Borel-Pompeiu formula (57) we may infer that C F α f = I − C T α RL D α a + g. In view of the continuity of C T α and the fact that for a function g ∈ W α,q a + (Ω) we have RL D α a + g ∈ W α,q a + (Ω), and hence we conclude that (I − C T αRL D α a + )g ∈ W α,q a + (Ω). By Theorem 4.6 and Theorem 4.7 we have 0 This in turn implies that g = C F α f ∈ W α,q a + (Ω) ∩ ker( C D α a + ) for a function g ∈ W α,q a + (Ω).

Hodge-type decomposition
The aim of this section is to obtain a Hodge-type decomposition and to present an immediate application of this decomposition for the resolution of boundary value problems involving the fractional Laplace operator. To realize this we need first the following lemma.
is given in the operator form by where the functions g 0 and g 1 are the Cauchy initial conditions given by Proof: The proof follows the same reasoning of the deduction of (33). Applying successively the fractional integrals I 1+αj a + j , with j = 1, . . . , n, to both sides of (65), applying Fubini's theorem, and rearranging the terms, we get where g 0 and g 1 are the Cauchy initial conditions given in (67). Applying the (n − 1)-dimensional Laplace transform with respect to x = (x 2 , . . . , x n ) to (68), taking into account the relations (24)- (28), and multiplying by n p=2 s 1+αp p we obtain the following second kind homogeneous integral equation of Volterra type: where G(x 1 , s) = G 0 ( s) + (x 1 − a 1 ) G 1 ( s) and G k ( s) = L {g k } (s) with k = 0, 1. Using (16), we have that the unique solution of (69) in the class of summable functions is: The first two terms in (70) coincide with (30) and are equal to (31) with λ = 0. Concerning the last two terms in (70), due the convergence of the integrals and the series, we can interchange them and rewrite them in the following way (in the calculations we make a change of the order of integration): It remains to invert the Laplace transform. Using (32) and after straightforward calculations, we obtain, which corresponds to our result.
Proof: By (− C ∆ α a + ) −1 0 we denote the unique operator solution for the problem (cf. Lemma 5.1) which is given by (66) with v(x) = f (x) and g 0 ( x) = 0. As a first step we take a look at the intersection of the two spaces that appear in the decomposition. Let f ∈ L q (Ω) ∩ ker( C D α a + ) ∩ C D α a + • W α,p a + (Ω) . We directly see that C D α a + f = 0, in Ω. Moreover, since f ∈ C D α a + • W α,p a + (Ω) , there exists a function g ∈ • W α,p a + (Ω) with C D α a + g = f and C ∆ α a + g = 0. From the uniqueness of (− C ∆ α a + ) −1 0 we obtain that g = 0. Consequently, f = 0 in Ω. Hence, the intersection of these subspaces only contains the zero function, which implies that the sum is direct. Now, let f ∈ L q (Ω) and f 2 such that Applying C D α a + to the function f 1 := f − f 2 , we get i.e., f 1 ∈ ker C D α a + . Since f ∈ L q (Ω) was arbitrarily chosen our decomposition is a direct decomposition of the space L q (Ω).
We end this section by presenting an immediate application of our results. is given by f = − C T α C Q α C T α g.

Proof:
The proof is based on applying the properties of the operator C T α and of the projector C Q α . Since C T α is the right inverse of C D α a + , we get C ∆ α a + f = C D α a + C D α a + C T α C Q α C T α g = C D α a + C Q α C T α g = C D α a + C T α g = g.

Conclusion
In this work we presented a generalization of several results of the classical continuous Clifford function theory developed in [17] in the context of fractional Clifford analysis. Our results can be regarded as a starting point for future works. Due to the "double duality" indicated previously, some of the previous results admit alternative versions, for instance, for the operator RL D α a + . Moreover, it is desirable to obtain an explicit expression for the fundamental solutions finding appropriate functions g 0 and g 1 . This can be done considering adequate series expansions in the neighbourhood of a. This will be subject for future work.