Abstract
In this paper boundary value problems for quaternionic Hermitian monogenic functions are presented using a circulant matrix approach.
MSC: 30G35.
Keywords:
quaternionic Clifford analysis; Cauchy integral formula1 Introduction
Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, i.e. null solutions of a firstorder vectorvalued rotation invariant differential operator, called Dirac operator, which factorises the Laplacian; monogenic functions may thus also be seen as a generalisation of holomorphic functions in the complex plane. Its roots go back to the 1930s. For more details on this function theory we refer to the standard references [5,12,1416].
More recently Hermitian Clifford analysis emerged as a refinement of the Euclidean setting for the case of . Here, Hermitian monogenic functions are considered, i.e. functions taking values either in a complex Clifford algebra or in complex spinor space, and being simultaneous null solutions of two complex Hermitian Dirac operators, which are invariant under the action of the unitary group. For the systematic development of this function theory we refer to [68].
In the papers [10,11,13,17], the Hermitian Clifford analysis setting was further refined by considering functions on with values in a quaternionic Clifford algebra, being simultaneous null solutions of four mutually related quaternionic Dirac operators, which are invariant under the action of the symplectic group. In [3], BorelPompeiu and Cauchy integral formulas are established in this setting, by following a () circulant matrix approach, similar in spirit to the circulant () matrix approach introduced in [9] within the complex Hermitian Clifford case. Subsequently, in [4] a quaternionic Hermitian Cauchy integral is introduced, as well as its boundary limit values, leading to the definition of a matrix quaternionic Hermitian Hilbert transform. These operators provide a useful tool for studying boundary value problems for the quaternionic Hermitian system. This is precisely the main objective of the present paper. The main problems that we address are the problem of finding a quaternionic Hermitian monogenic function with a given jump over a given surface of as well as problems of Dirichlet type for the quaternionic Hermitian system. Finally, we also prove an equivalence between bothsided quaternionic Hermitian monogenicity and a certain integral conservation law.
2 Preliminaries
Let be an orthonormal basis of Euclidean space and consider the real Clifford algebra constructed over . The noncommutative multiplication in is governed by the rules:
In one can consider the following automorphisms:
(i) the conjugation and for any ,
(ii) the main involution and for any , .
In particular, we consider the skewfield of quaternions whose elements will be denoted by with and . Clearly, may be identified with the Clifford algebra making the identifications , and . The automorphisms (i) and (ii) then respectively lead to the conjugation
However, it is quite natural to introduce two more involutions defined by
Definition 1 (see[17])
The quaternionic Witt basis of , , is given by , , where
We will consider the Clifford vectors
for which , while , , . The corresponding Dirac operators are denoted by , , and . Here we have , with the Laplacian in , and , , . Next, the quaternionic Hermitian variables are introduced:
for which , the symbol ^{†} denoting Hermitian quaternionic conjugation is defined as the composition of conjugation and Clifford conjugation in , i.e.. The Hermitian Dirac operators are
Definition 2 (see [17])
Let Ω be an open set in . A continuously differentiable function is said to be (left) qHermitian monogenic in Ω (or qmonogenic for short) iff it satisfies in Ω the system , or, equivalently, the system .
Similarly right qmonogenicity is defined. Left and right qmonogenic functions are called twosided qmonogenic. A qmonogenic function in Ω is monogenic, and thus harmonic in Ω. Note that Definition 2 was proven in [10] to be equivalent to the system introduced in [13] by group invariance considerations.
The fundamental solutions of the Dirac operators , , i.e. the Euclidean Cauchy kernels, are respectively given by
with the area of the unit sphere in . Explicitly, this means that , . Next we introduce the Hermitian Cauchy kernels:
Note that is not the fundamental solution of . However, the following theorem holds, see [3].
Theorem 1Introducing the circulant () matrices
Thus, is a fundamental solution of , in a matricial interpretation.
We associate, with functions , , and defined in and taking values in , the () circulant matrix function
We say that G belongs to some class of functions if all its entries belong to that class. In particular, the spaces of ktimes continuously differentiable, of αHölder continuous () and of pintegrable () circulant matrix functions on some suitable subset E of are respectively denoted by , and . The corresponding spaces of valued functions are denoted by , and . Moreover, introducing the nonnegative function , the classes and may also be defined by means of the respective traditional conditions
and
Definition 3 The () circulant matrix function G is called (left) QHermitian monogenic in Ω (or Qmonogenic for short) iff in Ω, where O denotes the matrix with zero entries.
Similarly right Qmonogenicity is defined by the system . Left and right Qmonogenic matrix functions are called twosided Qmonogenic. An important special case concerns the diagonal matrix function , with and . Indeed, is left (respectively right) Qmonogenic iff the function g is left (respectively right) qmonogenic.
Now, let be a bounded simply connected domain in with boundary , and denote by the complementary open domain . We assume Γ to be a Liapunov surface. The unit normal vector on Γ at is given by
and similarly as above, we also introduce the vectors , and , giving rise in the usual way (up to a constant factor) to their Hermitian counterparts
and , , , as well as to the circulant matrix . Then, in [3], the following Cauchy integral formulae were proven for Qmonogenic matrix functions and for qmonogenic functions, respectively.
Theorem 2 (QHermitian Cauchy integral formula)
If the matrix functionG, (1), isQmonogenic in Ω then
Theorem 3 (qHermitian Cauchy integral formula)
If the functiongisqmonogenic in Ω then
whereis the corresponding diagonal matrix.
Next, in [4] a QHermitian Cauchy transform was introduced, given by
for a matrix function , where and denote the Hermitian versions of the Clifford vectors and , respectively. is a left Qmonogenic matrix function in , vanishing at infinity; in terms of the Euclidean Cauchy type integrals
it reads as
In particular, for the special case of the matrix , the action of is reduced to
In general will not be a diagonal matrix, whence its entries will not be left qmonogenic functions. However does become diagonal if and only if
in which case we obtain
The following PlemeljSokhotski formula, proven in [4], then asserts the existence of the continuous boundary limits of the QHermitian Cauchy transform.
Theorem 4Let (), then the continuous limit values of itsQHermitian Cauchy transformexist and are given by
Here we have introduced the matrix QHermitian Hilbert operator
where the singular integrals
are Cauchy principal values. shows the following traditional properties, see [4].
Theorem 5One has
(i) is a bounded linear operator on ()
Similar results may be obtained for righthand versions of the QHermitian Cauchy and Hilbert transforms by means of the alternative definitions
and
where
3 Boundary value problems for Qmonogenic functions
In this section we study the socalled jump problem (reconstruction problem) for Qmonogenic functions; that is, we will investigate the problem of reconstructing a Qmonogenic matrix function Ψ in vanishing at infinity and having a prescribed jump G across Γ, i.e.
First, it should be noted that if this problem has a solution, then it necessarily is unique. This assertion can be easily proven using the Painlevé and Liouville theorems in the Clifford analysis setting, see [1]. Next, under the condition that , Theorem 4 ensures the solvability of the jump problem (4), its unique solution being given by
Now consider the important special case of the matrix function . The reconstruction problem (4) then is strongly related to the jump problem for the involved qmonogenic function, as addressed in the following theorem.
Theorem 6For a function, the following statements are equivalent:
(i) the jump problem
is solvable in terms ofqmonogenic functions;
(ii) gsatisfies the relations (3);
(iii) gsatisfies the relations.
Proof (i) → (ii)
Associate to the function g the diagonal matrix function . Then , and the jump problem (4) for has the unique solution
Let ψ be a solution of (5), then the circulant matrix
is another solution of the jump problem (4) for , whence the uniqueness yields
implying (ii).
(ii) → (iii)
From the third relation in (3), we have , and hence
the latter following from the second relation in (3) and the monogenicity of . This fact means that is a monogenic function in . Moreover, it has a null jump through Γ, whence it vanishes in the whole of . We conclude that . Similarly, we arrive at .
(iii) → (i)
It suffices to observe that, under the conditions stated, is qmonogenic, whence it solves the jump problem (5). □
For right qmonogenic functions the following analogue is obtained.
Theorem 7For a function, the following statements are equivalent:
(i) the jump problem
is solvable in terms of rightqmonogenic functions;
(ii) gsatisfies the relations
(iii) gsatisfies the relations.
The next result deals with the Dirichlet boundary value problem for Qmonogenic functions.
Theorem 8Let, then the following statements are equivalent:
(i) The Dirichlet problem
has a solution.
Proof We give the proof for the leftsided version of the theorem, the rightsided one being completely similar.
(i) → (ii)
Let F be a solution of the Dirichlet problem (7). Then, by the QHermitian Cauchy formula, we have
Taking limits as , (ii) follows in view of Theorem 4.
(ii) → (i)
It suffices to observe that, under the condition (ii), solves (7). □
Theorem 9Let, then the following statements are equivalent:
(i) The Dirichlet problem
has a solution.
(ii) gsatisfies the relations
(iii) gsatisfies the relations
Proof (i) → (ii)
From (i) we see that the matrix function
is a solution of the Dirichlet problem
whence by Theorem 8 we have that . The desired conclusion (ii) then directly follows by comparing the entries in the above equality.
(ii) → (iii)
From the condition it follows that . Therefore, as is harmonic in and , we have in . Using the remaining conditions in (ii) and following a similar reasoning as above, we obtain that g satisfies the relations (3) and hence by Theorem 6 we have that . Consequently, we obtain that , as stated in (iii).
(iii) → (i)
The conditions imply the solvability of the Dirichlet problems
where . Now, let , , , be the respective solutions of (9), then these functions all are solutions of the classical Dirichlet problem
whence they coincide. The function thus is qmonogenic and constitutes a solution of (8). □
For right qmonogenic functions the following analogue is obtained.
Theorem 10Let, then the following statements are equivalent:
(i) The Dirichlet problem
has a solution.
(ii) gsatisfies the relations
(iii) gsatisfies the relations
We now turn our attention towards establishing a connection between the twosided Qmonogenicity of a matrix function G and the matrix Hilbert transforms and of its trace on the boundary Γ.
Theorem 11Let, such thatin Ω, then the following statements are equivalent:
(i) Gis twosidedQmonogenic in Ω.
Proof Assume that, next to its already assumed left Qmonogenicity, G also is right Qmonogenic in Ω. Then by Theorem 8 it holds that
Conversely, suppose that . By Theorem 4 and its righthanded version, we conclude that the corresponding left and right QHermitian Cauchy transform of G, and , have the same boundary values on Γ. This fact, together with their harmonicity, implies that
On the other hand, from the assumed left Qmonogenicity of G we have and hence
which clearly forces G to be twosided Qmonogenic. □
The following result illustrates the utility of the above theorem when considering qmonogenic functions.
Theorem 12Letbe leftqmonogenic in Ω, then the following statements are equivalent:
(i) gis twosidedqmonogenic in Ω.
(ii) gsatisfies the relations
(iii) gsatisfies the relations
Proof (i) ↔ (ii)
From (i) we see that the matrix function corresponding to g is twosided Qmonogenic in Ω, whence (ii) follows from Theorem 11(ii) applied to . Conversely, (ii) can be rewritten in the matricial form , from which (i) follows by observing that the twosided Qmonogenicity of implied by Theorem 11 is equivalent to the qmonogenicity of g.
(i) ↔ (iii)
It follows from (i) that g is twosided monogenic w.r.t. , . We may then invoke [[2], Theorem 3.2] in order to conclude that , . Conversely, suppose that (iii) holds. Each of the conditions , , implies the twosided monogenicity of g in Ω w.r.t. , , see again [[2], Theorem 3.2], whence g is twosided qmonogenic in Ω. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors have worked jointly on the manuscript, which is the result of an intensive collaboration. All authors read and approved the final manuscript.
Acknowledgement
Ricardo AbreuBlaya and Juan BoryReyes wish to thank all members of the Department of Mathematical Analysis of Ghent University, where the paper was written, for the invitation and hospitality. They were supported respectively by the Research Council of Ghent University and by the Research Foundation  Flanders (FWO, project 31506208).
References

AbreuBlaya, R, BoryReyes, J, Peña Peña, D: Jump problem and removable singularities for monogenic functions. J. Geom. Anal.. 17(1), 1–13 (2007). PubMed Abstract  Publisher Full Text  PubMed Central Full Text

AbreuBlaya, R, BoryReyes, J, Delanghe, R, Sommen, F: Cauchy integral decomposition of multivector valued functions on hypersurfaces. Comput. Methods Funct. Theory. 5(1), 111–135 (2005)

AbreuBlaya, R, BoryReyes, J, Brackx, F, De Schepper, H, Sommen, F: Cauchy integral formulae in Hermitian quaternionic Clifford analysis. Complex Anal. Oper. Theory (accepted for publication). doi:10.1007/s1178501101688

AbreuBlaya, R, BoryReyes, J, Brackx, F, De Schepper, H, Sommen, F: Matrix Cauchy and Hilbert transforms in Hermitean quaternionic Clifford analysis. Complex Var. Elliptic Equ. (accepted for publication). doi:10.1080/17476933.2011.626034

Brackx, F, Delanghe, R, Sommen, F: Clifford Analysis, Pitman, Boston (1982)

Brackx, F, Bureš, J, De Schepper, H, Eelbode, D, Sommen, F, Souček, V: Fundaments of Hermitean Clifford analysis  part I: complex structure. Complex Anal. Oper. Theory. 1(3), 341–365 (2007). Publisher Full Text

Brackx, F, Bureš, J, De Schepper, H, Eelbode, D, Sommen, F, Souček, V: Fundaments of Hermitean Clifford analysis  part II: splitting of hmonogenic equations. Complex Var. Elliptic Equ.. 52(1011), 1063–1079 (2007). Publisher Full Text

Brackx, F, De Schepper, H, Sommen, F: The Hermitean Clifford analysis toolbox. Adv. Appl. Clifford Algebras. 18(34), 451–487 (2008). Publisher Full Text

Brackx, F, De Knock, B, De Schepper, H: A matrix Hilbert transform in Hermitean Clifford analysis. J. Math. Anal. Appl.. 344, 1068–1078 (2008). Publisher Full Text

Damiano, A, Eelbode, D, Sabadini, I: Quaternionic Hermitian spinor systems and compatibility conditions. Adv. Geom.. 11, 169–189 (2011)

Damiano, A, Eelbode, D, Sabadini, I: Algebraic analysis of Hermitian monogenic functions. C. R. Acad. Sci. Paris, Ser. I. 346, 139–142 (2008). Publisher Full Text

Delanghe, R, Sommen, F, Souček, V: Clifford Algebra and SpinorValued Functions, Kluwer Academic, Dordrecht (1992)

Eelbode, D: A Clifford algebraic framework for invariant differential operators. Adv. Appl. Clifford Algebras. 17, 635–649 (2007). Publisher Full Text

Gilbert, J, Murray, M: Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge (1991)

Gürlebeck, K, Habetha, K, Sprössig, W: Holomorphic Functions in the Plane and nDimensional Space, Birkhäuser, Basel (2008) Translated from the 2006 German original

Gürlebeck, K, Sprössig, W: Quaternionic and Clifford Calculus for Physicists and Engineers, Wiley, Chichester (1998)

PeñaPeña, D, Sabadini, I, Sommen, F: Quaternionic Clifford analysis: the Hermitian setting. Complex Anal. Oper. Theory. 1, 97–113 (2007). Publisher Full Text