# A higher dimensional fractional Borel‐Pompeiu formula and a related hypercomplex fractional operator calculus

Abstract: 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.

## Summary (2 min read)

### 1 Introduction

- Properties and applications of ∗The final version is published in Mathematical Methods in the Applied Science, in press.
- Additionally, in [4] the authors also investigated some interesting connections between the Teodorescu operator and Hermitian regular functions.
- 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).
- In Section 3, the authors present the fundamental solutions of the fractional Laplace and Dirac operators in Rn, defined by left Riemann-Liouville and Caputo fractional derivatives.

### 3 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.
- For the fractional Dirac operator CDαa+ the authors can obtain a family of fundamental solutions by applying the operator CDαa+ to the family of fundamental solutions of the operator C∆αa+ .
- Now the authors present the corresponding results for the Riemann-Liouville case.

### 4 Fractional Teodorescu and Cauchy-Bitsadze operators

- In this section the authors introduce and study the main properties of the fractional analogues of the classical Teodorescu and Cauchy-Bitsadze operators described in [17].
- Suppose that f and g satisfy the above mentioned conditions, also known as Proof.
- Before the authors deduce two properties of the fractional integral operators (59) and (60), they need to understand the behaviour of their fractional derivatives when the argument of the function over which they apply the derivatives is only translated.
- Concerning (63) the deduction is even more direct because the operator C a+k ∂ 1+αk 2 xk only acts on the functions g0 and g1.

### 5 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 the authors need first the following lemma.
- Hence, the intersection of these subspaces only contains the zero function, which implies that the sum is direct.
- Since f ∈ Lq(Ω) was arbitrarily chosen their decomposition is a direct decomposition of the space Lq(Ω).
- The proof is based on applying the properties of the operator CTα and of the projector CQα.

### 6 Conclusion

- In this work the authors presented a generalization of several results of the classical continuous Clifford function theory developed in [17] in the context of fractional Clifford analysis.
- Due to the “double duality” indicated previously, some of the previous results admit alternative versions, for instance, for the operator RLDαa+ .
- Moreover, it is desirable to obtain an explicit expression for the fundamental solutions finding appropriate functions g0 and g1.

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### "A higher dimensional fractional Bor..." refers background or methods in this paper

...By D a+ and D 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]) ( D a+f ) (x) = ( DmIm−α a+ f ) (x) = 1 Γ(m− α) d dxm ∫ x...

[...]

...Let D a+ and D b− denote, respectively, the left and right Caputo fractional derivative of order α > 0 on [a, b] ⊂ R, which are defined by (see [21]) ( D a+f ) (x) = ( Im−α a+ D f ) (x) = 1 Γ(m− α) ∫ x...

[...]

...Taking into account its convolution and operational properties (see [9,21]), we obtain the following relations for each term in (22):...

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...The left and right Riemann-Liouville fractional integrals I a+ and I α b− of order α are given by (see [21])...

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6,805 citations

### "A higher dimensional fractional Bor..." refers background in this paper

...4 in [26]), leads to ∥∥∥∥ C a+k 1+αk 2 xk (CTαg) ∥∥∥∥ Lq(Ω) = ∥∥∥∥( C a+k 1+αk 2 xk G + ) ∗ g ∥∥∥∥ Lq(Ω) ≤ ∥∥∥∥ C a+k 1+αk 2 xk G + ∥∥∥∥ L1(Ω) ‖g‖Lq(Ω)...

[...]

...If a function f admits a summable fractional derivative, then we have the following composition rules (see [26] and [25], respectively) ( I a+ D a+f ) (x) = f(x)− m−1 ∑...

[...]

...2 (see [26]) A function f ∈ L1(a, b) has a summable fractional derivative ( D a+f ) (x) if ( Im−α a+ f ) (x) belongs to AC([a, b]), where m = [α] + 1....

[...]

...[26]) A function f belongs to I a+(L1), α > 0, if and only if I m−α a+ f belongs to AC ([a, b]), m = [α] + 1 and (Im−α a+ f) (a) = 0, k = 0, ....

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...A description of this class of functions is given in [26]....

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3,556 citations

### "A higher dimensional fractional Bor..." refers methods in this paper

..., xn) to (68), taking into account the relations (24) to (28), and multiplying by ∏n p=2 s 1+αp p , we obtain the following second kind homogeneous integral equation of Volterra type:...

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2,819 citations

### "A higher dimensional fractional Bor..." refers background in this paper

...If a function f admits a summable fractional derivative, then we have the following composition rules (see [26] and [25], respectively) ( I a+ D a+f ) (x) = f(x)− m−1 ∑...

[...]