Let be the space of -vectors in the Clifford
algebra constructed over the quadratic vector space , let with , and , and let . Then, an -valued smooth function defined in an open subset is said to satisfy the generalized
Moisil-Théodoresco system of type if in , where is the Dirac operator in . A structure theorem is proved
for such functions, based on the construction of conjugate harmonic
pairs. Furthermore, if is bounded with boundary , where is
an Ahlfors-David regular surface, and if is a -valued Hölder
continuous function on , then necessary and sufficient conditions are
given under which admits on a Cauchy integral decomposition
.
1. Introduction
Clifford analysis, a function theory for the Dirac
operator in Euclidean space (), generalizes in an elegant way the theory of
holomorphic functions in the complex plane to a higher dimension and provides
at the same time a refinement of the theory of harmonic functions. One of the
basic properties relied upon in building up this function theory is the fact
that the Dirac operator in factorizes the
Laplacian through the
relation . The Dirac operator is defined by , where and is an
orthogonal basis for the quadratic space , the latter being the space equipped with a
quadratic form of signature . By virtue of the basic multiplication rulesvalid in the universal Clifford
algebra constructed
over , the factorization is thus
obtained.
Notice that is a real
linear associative algebra of dimension , having as standard basis the set
where
, , , and , the identity element in .
Now let be open and let be a -function in . Then, is said to be
left monogenic in if in . The equation gives rise to a
first-order linear elliptic system of partial differential equations in the
components of By choosing as an
orthogonal basis for the quadratic space , then inside , thus generates
the Clifford algebra . It is then easily seen thatwhere . If the -valued -function in is decomposed
following (1.2), that is where and are -valued -functions in , then in where is the
Cauchy-Riemann operator in and is the Dirac
operator in .
Obviously the system (1.3) generalizes the classical
Cauchy-Riemann system in the plane: it indeed suffices in the case to take -valued and -valued.
Left monogenic functions in are real
analytic, whence by virtue of , they are in particular -valued and
harmonic in .
As the algebra is
noncommutative, one could as well consider right monogenic functions in , that is satisfies the
equation in . If both and in , then is said to be
two-sided monogenic in .
Notice also that through a natural linear isomorphism (see Section 2), the spaces and of smooth -valued
functions and smooth differential forms in may be
identified. The left and right actions of on then correspond
to the actions of and on , where and denote,
respectively, the exterior derivative and the coderivative operators. For the
sake of completeness, let us recall the definition of and on the space of smooth -forms in , (see [1]).
For with where , , and are defined
by A smooth differential form satisfying in was called in
[2] a self-conjugate
differential form.
It thus becomes clear that through the identifications
mentioned (see again Section 2) a subsystem of (1.3) corresponds to a subsystem
of self-conjugate differential forms and vice versa. For instance, for fixed, the
study of left monogenic -vector valued
functions thus
corresponds to the study of -forms satisfying the
Hodge-de Rham system and .
Let us recall that the space of -vectors in () is defined byFor an account on recent
investigations on subsystems of (1.3) or, equivalently, on the study of
particular systems of self-conjugate differential forms, we refer to [2–10].
Now fix , take such that and , and putThe present paper is devoted to
the study of -valued smooth
functions in which are left
monogenic in (i.e., which
satisfy in ). The space of
such functions is henceforth denoted by . The system defines a
subsystem of (1.3), called the generalized Moisil-Théodoresco system of type in .
To be more precise, let us first recall the definition
of the differential operators and acting on
smooth -valued
functions in . Call the space of
smooth -valued
functions in and put for ,
Note that is -valued while is -valued and
that through the isomorphism , the action of and on corresponds to,
respectively, the action of and on the space .
If is written
aswe then have that the
generalized Moisil-Théodoresco system of type reads as
follows (see also Section 2):Note that for and fixed, the
system (1.9) reduces to the generalized Riesz system Its solutions are called harmonic multivector fields (see also
[11]). We
haveFurthermore, for , and fixed, the
system (1.9) reduces to the Moisil-Théodoreco system in (see, e.g.,
[3]):In the particular case, where , and , the original Moisil-Théodoresco system introduced in
[12] is reobtained
(see also [4]).
In this paper, two problems are dealt with;
we list them as follows.
(i)To characterize
the structure of solutions to the system (1.9).
It is proved in
Section 4 (see Theorem 3.2) that, under certain geometric conditions upon , each corresponds to
a harmonic potential belonging to a
particular subspace of the space of harmonic -valued
functions in .
The proof of Theorem 3.2 relies heavily on the
construction of conjugate harmonic pairs elaborated in Section 3.
(ii)To characterize those which admit a
Cauchy-type integral decomposition on of the
form
where is the boundary
of a bounded open domain in and denotes the
space of -valued Hölder
continuous functions of order on , . Putting , the elements and should also
belong to and as such
should be the boundary values of solutions and of (1.9) in and , respectively.
In Section 5, this problem is solved in terms of the
Cauchy transform on , being an -dimensional
Ahlfors-David regular surface (see Theorem 4.2).
In order to make the paper self-contained, we include
in Section 2 some basic properties of Clifford algebras and Clifford analysis.
For a general account of this function theory, we refer, for example, to the
monographs [13–15].
2. Clifford Analysis: Notations and Some Basic Properties
Let again be an
orthogonal basis for and let be the universal
Clifford algebra over . As has already been mentioned in Section 1, is a real
linear associative but noncommutative algebra of dimension ; its standard basis is given by the set and the basic
multiplication rules are governed by (1.1). For fixed, the
space of -vectors is
defined by (1.5), leading to the decompositionand the associated projection
operators .
Note in particular that for , and that for , .
An element is therefore
usually identified with .
For , the product splits in two
parts, namely,where is the scalar
part of and is the 2-vector
or bivector part of . They are given by
More generally, for and (), we have that the product decomposes
intowhere
Another useful decomposition of may be obtained
by splitting it “along the -direction,”
as indicated in (1.2). This in fact means that we split following and that within , the Clifford algebra is generated by
the orthogonal basis of . denotes the
space to which the
original quadratic form of signature on has been
restricted.
Following the decomposition (1.2), the element is then often
identified with the so-called paravector .
Let us also recall that if is open and is an -valued -function in , then is said to be
left monogenic in if in , being the Dirac
operator in .
As already mentioned in (1.3), by putting , being the Dirac
operator in , we have for ,Let us recall that a pair of -valued
harmonic functions in is said to be
conjugate harmonic if is left
monogenic in (see [16]).
Notice also that, when defining the conjugate of by , we have that .
If is a subspace
of , then and denote,
respectively, the spaces of left monogenic and harmonic -valued
functions in . As we have that .
In particular, for such that , with , we have put in Section 1 (see (1.6)), and .
Furthermore, for fixed, a
natural isomorphismmay be then defined as follows.
Put for ,where for each with , and for all .
By means of the decomposition (2.1), may be extended
by linearity to , thus leading to the isomorphism , where as usual .
It may be easily checked that the action of the
exterior derivative and the
co-derivative on then
corresponds through to the left
action of and on . For the definition of and (resp., and ) we refer to (1.4) and (1.7). In fact, taking into account the relations (2.5), the expressions (1.7) mean that for ,
Consequently, for , splits
intoIt thus follows that for , the system is given by
(1.9).
Obviously, for , , while for , . Finally, notice that and that hence,
as mentioned in Section 1, through , the left action of on corresponds to
the action of on . We thus have on that .
The following notations will also be
used:
Let us recall that if is contractible
to a point, a refined version of the inverse Poincaré lemma then implies
that
are surjective operators.
For the inverse Poincaré lemma and its refined version
we refer to, respectively, [1, 17]. For more information concerning the interplay
between differential forms and multivectors, the reader is referred to
[17, 18].
Obviously, all notions, notations, and properties
introduced above may be easily adapted to the case where is the
orthogonal projection of on and and are the Dirac
and Laplace operators in .
3. Conjugate Harmonic Pairs
Let be as in
Section 1, let with , and decompose each following (1.2),
that iswhere and .
Then, with
Now suppose that , that is, is a conjugate
harmonic pair in in the sense of
[16]. Then, as already
stated in (1.3),By virtue of (2.10) and (3.2), the
equations in (3.3) lead to the systemsFrom (3.5) it thus follows that implies that in .
We now claim that, under certain geometric conditions
upon , given , harmonic and -valued in , the condition in is sufficient
to ensure the existence of a , harmonic and -valued in , which is conjugate harmonic to , that is .
In proving this statement, we will adapt where
necessary the techniques worked out in [16] for constructing conjugate harmonic pairs.
Let again denote the orthogonal
projection of on . Then, we suppose henceforth that satisfies the
following conditions (C1) and (C2):
(C1) is normal with
respect to the direction, that
is, there exists such that for
all , is connected
and it contains the element ;(C2) is contractible
to a point.
The condition (C1) is sufficient
for constructing harmonic conjugates to (see [16]), while the condition (C2) ensures the
applicability of the inverse Poincaré lemma and its consequences in (see [17]).
As is well known, classical results of cohomology
theory provide necessary and sufficient conditions for the validity of the
inverse Poincaré lemma in . For convenience of the reader, we restrict ourselves
to the condition (C2), thus making the inverse Poincaré lemma applicable
for any closed or coclosed form in ().
Now assume that harmonic and
that -valued in satisfies the
condition in .
Putwhere is a smooth -valued
solution in of the
equationAs is surjective
(see [19]), such indeed exists
and any other similar solution of (3.7) has the form where .
Fix a solution of (3.7). Then
by construction, the corresponding determined by
(3.6) belongs to (see [16]).
We now prove that there exists such that in ,To this end, first notice that,
as by assumption in , we have that in , whence .
As is surjective (see also (2.12)) there exists such that , that is, satisfies in the
relationsFurthermore, put . Then, on the one hand,while on the other
handConsequently, in and is -valued in .
Now define byThen by construction, and clearly in , .
Furthermore, as in , is -valued and
obviously is -valued. As
moreover , we get that is -valued.
Consequently, as , , that is, is a conjugate
harmonic pair in .
We have thus proved the following
theorem.
Theorem 3.1. Let be open and
normal with respect to the -direction and
let be contractible
to a point. Furthermore, let be given. Then, admits a
conjugate harmonic if and only if in .
Remarks
(1) If , then in , thus implying that the condition is
automatically satisfied and that in constructing , no correction term should be added
to (i.e., we may
take in
(3.12)).
(2) It is of course
tacitly understood that if , then in the expression of (see (3.2)), the
first term is taken to be
identically zero in .
(3) The systems (3.4) and (3.5) show a lot of symmetry.
The following
theorem (Theorem 3.2) holds, the proof of which is omitted.
Theorem 3.2. Let be open and
normal with respect to the -direction and
let be contractible
to a point. Furthermore, let be given. Then, admits a
conjugate harmonic if and only if in .
4. Structure Theorems
Assume that are such that and that with .
This section essentially deals with the construction
of harmonic potentials corresponding to solutions of the generalized
Moisil-Théodoresco system.
We start with the following
lemma.
Lemma 4.1. Let be open and
contractible to a point and let . The following properties are equivalent:
(i),(ii)there exists such that .
Proof. It is clear that if , then is -valued. As
moreover , ; whence (ii)⇒(i) is proved.
Conversely, assume that and put . From it follows
thatBy a refined version of the
inverse Poincaré lemma (see [17]) we obtain from the first equation in (4.1) that there
exists such that in Analogously, the third equation
in (4.1) implies the existence ofsuch that in Put . Then,and by virtue of (4.2) and (4.4), But where is -valued and
harmonic in . As is surjective
(see [19]), there
exists that such that .
Put . Then, clearly andFinally, put . Then, is -valued
andAs is obviously
harmonic in , the proof is done.
Remarks
(1) In the case
where and , we have that in (4.1) the equation is
automatically satisfied. Putting , take such that and define by . Then, .
In the case where , the equation is
automatically satisfied. An analogous reasoning to the one just made then leads
to an appropriate such that .
(2) Obviously, in
the case where as well as , the technique
suggested in Remark (1) then produces such that .
(3) A particularly important
example where as well as occurs when . Indeed, put for given real valued smooth functions in , ,Then, for ,Both equations in (4.10) give rise
to the same system to be satisfied by , namelyThe system (4.11) is the Fueter
system in for so-called
left regular functions of a quaternion variable; it lies at the basis of
quaternionic analysis (see [20, 21]).
We have taken this example from [2], where it was proved in the
framework of self-conjugate differential forms. We have inserted it here
because it demonstrates how quaternionic analysis can be viewed upon as part of
Clifford analysis in , namely as the theory of special solutions to a
generalized Moisil-Théodoresco system in of type .
(4) In the case
where and , Lemma 4.1 tells us that, given , there exists such that . This result was already obtained in [3, Lemma 3.1].
Theorem 4.2. Let be open and
normal with respect to the -direction, let be contractible
to a point, and let . The following properties are equivalent:
(i),(ii)there exists with in such that .
Proof. (i)→(ii). Let and put,
following (1.2), . Then, the pair is conjugate
harmonic in with and .
Associate with the harmonic -valued
potential given by (3.12),
that is,where in , , and .
As moreover in (see Theorem
3.1), it thus follows from (4.12) that in . Consequently, with and .
From , it is then easily obtained that is independent
of and that in , , that is, . By virtue of Lemma 4.1, there exists such that ; whence .
Put . Then by construction,
(i),(ii),(iii);
whence (i)→(ii) is proved.
Conversely, let with . Then clearly .
Remarks
(1) Theorem 4.2 tells
us that each admits an - valued
harmonic potential in satisfying .
(2) Let , that is, we take and . Then from Theorem 4.2, it follows that the following
properties are equivalent:
(i),(ii)there
exists with such that .
This
characterization was already obtained in [3, Theorem 3.1].
5. Cauchy Integral Decompositions
Let be a bounded
open subset with boundary where is a
rectifiable closed Jordan curve such that for some constant and this for
all and , where is the closed
disc with center and radius and is the
1-dimensional Hausdorff measure on . Furthermore, let and let , .
In classical complex analysis, the following jump
problem (5.1) is solved by means of the Cauchy transform:
“Find a pair of functions and , holomorphic in and with , such that are
continuously extendable to and that on where in (5.1), .”
Let be the Cauchy
transform on , that is, for ,where is the outward
pointing unit normal at and is the
elementary Lebesgue measure on .
Then, the following fundamental properties hold (see,
e.g., [22]):
(i) is holomorphic
and of the class on with ;(ii)Plemelj-Sokhotzki formulae:where for ,define the Hilbert transform on ;(iii) on .
It thus follows
that the answer to the jump problem (5.1) is indeed given by .
The decomposition (iii) thus obtained is known as the
Cauchy integral decomposition of on .
Now let be a bounded
and open subset of with boundary . Then, in Clifford analysis, for suitable pairs of boundaries and -valued
functions on , the Cauchy transform is defined
by
where the following conditions hold.
(i), , is the fundamental solution of the Dirac operator , where is the area of
the unit sphere in . is -valued and
monogenic in .(ii) is the outward
pointing unit normal at .(iii) is the -dimensional
Hausdorff measure on . For the definition of , see, for example, [23, 24].
In what follows
we restrict ourselves to the following conditions on the pair (see also the
remarks made at the end of this section).
(C1) is an -dimensional
Ahlfors-David regular surface, that is, there exists a constant such that for
all and ,where is the closed
ball in with center and radius and is the diameter
of .For the definition of AD-regular surfaces, see, for
example, [24, 25].(C2), , being the space
of -valued Hölder
continuous functions of order on .
Under the
conditions (C1) and (C2), the following properties hold (see, e.g.,
[26–28]):
(i) is left
monogenic in and ;(ii)(Plemelj-Sokhotzki formulae) the functions determined
bybelong to , wherethe integral being taken in the
sense of principal values;(iii), .
It thus follows
that, given a Hölder continuous -valued density on , the jump problem (5.9)
“Find and , belonging to and which are
the boundary values of left monogenic functions and in,
respectively, and with such that on
is solved by considering the
Cauchy transform . Indeed, we can take in and in .
Now let again be a triplet
satisfying and with , and let .
As is -valued, it is
easily seen that is -valued.
Consequently, if the jump problem (5.9) is formulated in terms of -valued Hölder
continuous functions , , and on , then if we wish to solve it by means of the Cauchy
transform , restrictions on have to be
imposed, namely, in we should
have
The very heart of the following
theorem (Theorem 5.1) tells us that the conditions (5.10) are necessary and
sufficient. Although the arguments used in proving Theorem 5.1 are similar to the
ones given in the proof of [29, Theorem 4.1], for convenience of the reader we write
them out in full detail.
Theorem 5.1. Let be open and
bounded such that is an -dimensional
Ahlfors-David regular surface and let with and . The following properties are equivalent:
(i) admits on a decomposition , where belong to and moreover
are the boundary values of functions with ,(ii),(iii),(iv) and in .
Proof. (i)→(ii). Assume that , where and satisfy the
conditions given in (i). ThenIn view of the assumptions made
on , we have that in , in and that in .
Consequently,As with
(ii) is
proved.
(ii)→(iii): Trivial.
(iii)→(iv). Let us first recall that is left
monogenic in with .
According to the decompositionand by the assumption made on , it follows from [1] thatFurthermore, as and split into an and an , respectively, into an and an multivector, we
obtain from [22] that
in
Moreover, as by assumption is -valued, by
virtue of the Plemelj-Sokhotzki formulae, we obtain that on
Furthermore, and are Hölder
continuous on .
It thus follows that and are left
monogenic in and
continuously extendable to . Painlevé's theorem (see [30]) then implies that and are left
monogenic in .
Finally, as , we obtain by virtue of Liouville's theorem (see
[31]) that and in .
(iv)→(i). First note that, as , by means of the Plemelj-Sokhotzki formulae, we have
on thatIn view of the assumption (iv) made, the functions defined in by obviously belong to and they
satisfy all required properties.
Remarks
(1) In the last
decades, intensive research has been done in studying the Cauchy integral
transform and the associated singular integral operator on curves in the plane or
on hypersurfaces in . Two types of boundary data are usually considered,
namely a Hölder continuous density or an -density .
In this section, we have formulated the jump problems
(5.1) and (5.9) in terms of Hölder continuous densities. The reason for this is
that in proving some of the equivalences stated in Theorem 5.1, the continuous
extendability of the Cauchy integral up to the boundary plays a crucial role.
This becomes clear for instance when use is made of Painlevé's theorem in
proving the implication “(iii)→(iv)”.
Note that for (), the continuous extendability of was already obtained
in 1965 by V. Iftimie in the case where is a compact
Liapunov surface (see [32]). For an overview of recent investigations on
conditions which can be put on the pair , being a
continuous density on , we refer the reader to [28, 30, 33–38]. In particular, we wish to point out that the
introduction and the references in [35] contain a detailed account of the historical
background of the jump problems (5.1) and (5.9).
(2) The case and was dealt with
in [39]. For open, bounded
and connected with -boundary such that is also
connected, a set of equivalent properties was obtained ensuring the validity of
the Cauchy integral decomposition for given.
(3) If wherethen , that is the condition (ii) in Theorem 5.1 is
satisfied.
Analogously, if wherethen and so the
condition (ii) in Theorem 5.1 is again satisfied.
Acknowledgments
The central idea for this paper arose while the second
author was visiting the Department of
Mathematical Analysis of Ghent University. He was supported by the Special
Research Fund no. 01T13804 of Ghent University obtained for collaboration
between the Clifford Research Group in Ghent and the Cuban Research Group in
Clifford analysis, on the subject Boundary value theory in Clifford Analysis.
Juan Bory Reyes wishes to thank the members of this Department for their kind
hospitality. The authors also are much grateful to
the referees: their questions and remarks contributed to improve substantially
the final presentation of this paper.