A higher dimensional fractional Borel-Pompeiu formula
and a related hypercomplex fractional operator calculus
∗
M. Ferreira
§,‡
, R.S. Kraußhar
]
, M.M. Rodrigues
‡
, N. Vieira
‡
§
School of Technology and Management,
Polytechnic Institute of Leiria
P-2411-901, Leiria, Portugal.
E-mail: milton.ferreira@ipleiria.pt
]
Fachgebiet Mathematik,
Erziehungswissenschaftliche Fakult¨at, Universit¨at Erfurt
Nordh¨auserstr. 63, 99089 Erfurt.
E-mail: soeren.krausshar@uni-erfurt.de
‡
CIDMA - Center for Research and Development in Mathematics and Applications
Department of Mathematics, University of Aveiro
Campus Universit´ario de Santiago, 3810-193 Aveiro, Portugal.
E-mails: mferreira@ua.pt, mrodrigues@ua.pt, nloureirovieira@gmail.com
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 decom-
position 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.
Keywords: Fractional Clifford analysis; Fractional derivatives; Stokes’s formula; Borel-Pompeiu for-
mula; Cauchy’s integral formula; Hodge-type decomposition.
MSC 2010: 35R11; 30G35; 26A33; 35A08; 30E20; 45P05.
1 Introduction
Clifford analysis offers a higher dimensional generalization of the classical theory of complex holomorphic func-
tions. 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 final version is published in Mathematical Methods in the Applied Science, in press. It as available via the website: ???.
1
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. G¨urlebeck and W. Spr¨oßig 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 Schr¨odinger 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 calcu-
lus 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 [10–12, 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 so-
lutions 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
2
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.
2 Preliminaries
2.1 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])
(I
α
a
+
f) (x) =
1
Γ(α)
Z
x
a
f(t)
(x −t)
1−α
dt, x > a (1)
(I
α
b
−
f) (x) =
1
Γ(α)
Z
b
x
f(t)
(t −x)
1−α
dt, x < b. (2)
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])
RL
D
α
a
+
f
(x) =
D
m
I
m−α
a
+
f
(x) =
1
Γ(m −α)
d
m
dx
m
Z
x
a
f(t)
(x −t)
α−m+1
dt, x > a (3)
RL
D
α
b
−
f
(x) = (−1)
m
D
m
I
m−α
b
−
f
(x) =
(−1)
m
Γ(m −α)
d
m
dx
m
Z
b
x
f(t)
(t −x)
α−m+1
dt, x < b. (4)
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])
C
D
α
a
+
f
(x) =
I
m−α
a
+
D
m
f
(x) =
1
Γ(m −α)
Z
x
a
f
(m)
(t)
(x −t)
α−m+1
dt, x > a (5)
C
D
α
b
−
f
(x) = (−1)
m
I
m−α
b
−
D
m
f
(x) =
(−1)
m
Γ(m −α)
Z
b
x
f
(m)
(t)
(t −x)
α−m+1
dt, x < b. (6)
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].
Theorem 2.1 (cf. [26]) A function f belongs to I
α
a
+
(L
1
), α > 0, if and only if I
m−α
a
+
f belongs to AC
m
([a, b]),
m = [α] + 1 and (I
m−α
a
+
f)
(k)
(a) = 0, k = 0, . . . , m −1.
In Theorem 2.1, AC
m
([a, b]) denotes the class of functions f which are continuously differentiable on the segment
[a, b] up to the order m − 1 and f
(m−1)
is supposed to be absolutely continuous on [a, b]. We note that the
conditions (I
m−α
a
+
f)
(k)
(a) = 0, k = 0, . . . , m−1, imply that f
(k)
(a) = 0, k = 0, . . . , m−1 (see [25,26]). Removing
the last condition in Theorem 2.1 we obtain the class of functions that admit a summable fractional derivative.
Definition 2.2 (see [26]) A function f ∈ L
1
(a, b) has a summable fractional derivative
D
α
a
+
f
(x) if
I
m−α
a
+
f
(x)
belongs to AC
m
([a, b]), where m = [α] + 1.
If a function f admits a summable fractional derivative, then we have the following composition rules (see [26]
and [25], respectively)
I
α
a
+
RL
D
α
a
+
f
(x) = f(x) −
m−1
X
k=0
(x −a)
α−k−1
Γ(α − k)
I
m−α
a
+
f
(m−k−1)
(a), m = [α] + 1 (7)
I
α
a
+
C
D
α
a
+
f
(x) = f(x) −
m−1
X
k=0
f
(k)
(a)
k!
(x −a)
k
, m = [α] + 1. (8)
3
We remark that if f ∈ I
α
a
+
(L
1
) then (7) and (8) reduce to
I
α
a
+
RL
D
α
a
+
f
(x) =
I
α
a
+
C
D
α
a
+
f
(x) = f(x). Nev-
ertheless we note that D
α
a
+
I
α
a
+
f = f in both cases. This is a particular case of a more general property
(cf. [25, (2.114)])
D
α
a
+
I
γ
a
+
f
= D
α−γ
a
+
f, α ≥ γ > 0. (9)
One important function used in this paper is the two-parameter Mittag-Leffler function E
µ,ν
(z) (see [16]),
which is defined in terms of the power series by
E
µ,ν
(z) =
∞
X
n=0
z
n
Γ(µn + ν)
, µ > 0, ν > 0, z ∈ C. (10)
In particular, the function E
µ,ν
(z) is entire of order ρ =
1
µ
and type σ = 1. From the power series (10) and the
operators (1), (3) and (5), we can obtain by straightforward calculations the following fractional integral and
differential formulae involving E
µ,ν
(z) (see [16, pp. 87-88]):
I
α
a
+
(x −a)
ν−1
E
µ,ν
(k(x − a)
µ
)
= (x −a)
α+ν−1
E
µ,ν+α
(k(x − a)
µ
) (11)
for all α > 0, k ∈ C, x > a, µ > 0, ν > 0,
RL
D
α
a
+
(x −a)
ν−1
E
µ,ν
(k(x − a)
µ
)
= (x −a)
ν−α−1
E
µ,ν−α
(k(x − a)
µ
) (12)
for all α > 0, k ∈ C, x > a, µ > 0, ν > 0, ν 6= α −p, where p = 0, . . . , m − 1 with m = [α] + 1, and
C
D
α
a
+
(x − a)
ν−1
E
µ,ν
(k(x − a)
µ
)
= (x − a)
ν−α−1
E
µ,ν−α
(k(x − a)
µ
) (13)
for all α > 0, k ∈ C, x > a, µ > 0, ν > 0, ν 6= p, where p = 1, . . . , m with m = [α] + 1.
Remark 2.3 For ν = α − p with p = 0, . . . , m − 1, we have that
RL
D
α
a
+
((x − a)
α−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 (12)
must be replaced in these cases by the following derivation rule:
RL
D
α
a
+
(x − a)
α−p−1
E
µ,α−p
(k(x − a)
µ
)
= (x − a)
µ−p−1
k E
µ,µ−p
(k(x − a)
µ
) , p = 0, . . . , m − 1. (14)
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:
C
D
α
a
+
(x − a)
p−1
E
µ,p
(k(x − a)
µ
)
= (x − a)
µ+p−α−1
k E
µ,µ+p−α
(k(x − a)
µ
) , p = 1, . . . , m. (15)
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.
Theorem 2.5 ( [16, Thm. 4.2]) Let f ∈ L
1
[a, b], α > 0 and λ ∈ C. Then the integral equation
u(x) = f(x) +
λ
Γ(α)
Z
x
a
(x − t)
α−1
u(t) dt, x ∈ [a, b]
has a unique solution
u(x) = f(x) + λ
Z
x
a
(x − t)
α−1
E
α,α
(λ(x − t)
α
)f(t) dt. (16)
Now we recall the formula for fractional integration by parts for 0 < α < 1 and x ∈ [a, b] (see [1])
Z
b
a
g(x)
C
D
α
a
+
f
(x) dx =
Z
b
a
f(x)
RL
D
α
b
−
g
(x) dx + [f(x) (I
α
b
−
g) (x)]
b
a
,
Z
b
a
g(x)
C
D
α
b
−
f
(x) dx =
Z
b
a
f(x)
RL
D
α
a
+
g
(x) dx − [f(x) I
α
a
+
g(x)]
b
a
.
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 I
α
b
−
are bounded from L
p
(a, b) into
L
q
(a, b), where q =
p
1−αp
and [a, b] ⊂ R.
4
2.2 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
e
i
e
j
+ e
j
e
i
= −2δ
ij
, (17)
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
l
1
e
l
2
. . . e
l
r
, 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 =
P
A
a
A
e
A
, with a
A
∈ R. The conjugation in the Clifford algebra R
0,n
is defined by a =
P
A
a
A
e
A
,
where e
A
= e
l
r
e
l
r−1
. . . e
l
1
, 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 =
P
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 =
P
n
j=1
e
j
∂
x
j
. This operator satisfies D
2
= −∆, where ∆ is the n-dimensional
Euclidean Laplacian. An R
0,n
-valued function f is called left-monogenic if it satisfies Du = 0 on Ω (resp.
right-monogenic if it satisfies uD = 0 on Ω).
For more details about Clifford algebras and basic concepts of its associated function theory we refer the
interested reader for example to [8].
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. In this section we present the generalization of
these results for R
n
. Let Ω =
Q
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
RL
∆
α
a
+
=
n
X
i=1
RL
a
+
i
∂
1+α
i
x
i
,
C
∆
α
a
+
=
n
X
i=1
C
a
+
i
∂
1+α
i
x
i
. (18)
Associated to them there are the corresponding fractional Dirac operators
RL
D
α
a
+
and
C
D
α
a
+
defined by
RL
D
α
a
+
=
n
X
i=1
e
i
RL
a
+
i
∂
1+α
i
2
x
i
,
C
D
α
a
+
=
n
X
i=1
e
i
C
a
+
i
∂
1+α
i
2
x
i
. (19)
For i = 1, . . . , n the partial derivatives
RL
a
+
i
∂
1+α
i
x
i
,
RL
a
+
i
∂
1+α
i
2
x
i
,
C
a
+
i
∂
1+α
i
x
i
and
C
a
+
i
∂
1+α
i
2
x
i
are the left Riemann-Liouville
and Caputo fractional derivatives (3) and (5) of orders 1 + α
i
and
1+α
i
2
, with respect to the variable x
i
∈]a
i
, b
i
[.
Under certain conditions we have that
RL
∆
α
a
+
= −
RL
D
α
a
+
RL
D
α
a
+
(see [10]), and
C
∆
α
a
+
= −
C
D
α
a
+
C
D
α
a
+
(see
[12]). Due to the nature of the eigenfunctions and the fundamental solution of these operators we additionally
need to consider the variable bx = (x
2
, . . . , x
n
) ∈
b
Ω =
Q
n
i=2
]a
i
, b
i
[, and the fractional Laplace and Dirac operators
acting on bx defined by
RL
b
∆
α
a
+
=
n
X
i=2
RL
a
+
i
∂
1+α
i
x
i
,
C
b
∆
α
a
+
=
n
X
i=2
C
a
+
i
∂
1+α
i
x
i
,
RL
b
D
α
a
+
=
n
X
i=2
e
i
RL
a
+
i
∂
1+α
i
2
x
i
,
C
b
D
α
a
+
=
n
X
i=2
e
i
C
a
+
i
∂
1+α
i
2
x
i
. (20)
We start by addressing the Caputo case. Consider the eigenfunction problem
C
∆
α
a
+
v(x) = λv(x), (21)
5