Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions of two Hermitian Dirac operators. Using a circulant matrix approach, we will study the Riemann type problems in Hermitian Clifford analysis. We prove a mean value formula for the Hermitian monogenic function. We obtain a Liouville-type theorem and a maximum module for the function above. Applying the Plemelj formula, integral representation formulas, and a Liouville-type theorem, we prove that the Riemann type problems for Hermitian monogenic and Hermitian-2-monogenic functions are solvable. Explicit representation formulas of the solutions are also given.
Keywords:Hermitian Clifford analysis; Riemann type problems; Hermitian monogenic function
The classical Riemann boundary value problem (BVP for short) theory in the complex plane has been systematically developed, see  and . It is natural to generalize the classical Riemann BVP theory to higher dimensions. Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis and a generation of complex in plane analysis. The theory is centered around the concept of monogenic functions, see [3-6], etc. Under the framework, in [7-12], many interesting results about BVP for monogenic functions in Clifford analysis were presented. In  and , Riemann BVP for harmonic functions (i.e., 2-monogenic functions) and biharmonic functions were studied, the solutions are given in an explicit way.
More recently, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions of two Hermitian Dirac operators invariant under the action of the unitary group. This function theory can be found in  and , etc. In , based on the complex Clifford algebra , the Hermitian Cauchy integral formulas were constructed in the framework of circulant matrix functions, and the intimate relationship with holomorphic function theory of several complex variables was considered. For details, we refer to [17-20]. In  and , a matrix Hilbert transform in Hermitian Clifford analysis was studied, and analogs of characteristic properties of the matrix Hilbert transform in classical analysis and orthogonal Clifford analysis were given, for example by the usual Plemelj-Sokhotski formula. Under this setting it is natural to consider the Riemann BVP. In , the Riemann BVP for (left) Helmholtz H-monogenic functions (i.e., null solutions of perturbed Hermitian Dirac operators in the framework of Hermitian Clifford analysis). If the perturbed value vanishes, is , then the Riemann BVP for H-monogenic circulant matrix functions was solved. Also, we naturally consider Riemann BVP for H-monogenic circulant matrix functions (i.e., null solutions to ) and H-2-monogenic circulant matrix functions (i.e., null solutions to ). Roughly speaking Riemann BVP means that we prescribe that the solutions are bounded at infinity. Up to present, as far as we know, it is a new problem. In this paper, motivated by [8,9,13,14,17,18], we will consider Riemann BVP for H-2-monogenic circulant matrix functions in Hermitian Clifford analysis. Applying the integral representation formulas of H-monogenic circulant matrix functions and H-2-monogenic circulant matrix functions, we get mean values formulas. Furthermore we prove a maximum modulus theorem and a Liouville theorem in Hermitian Clifford analysis. Finally we get explicit solutions for Riemann BVP for H-2-monogenic circulant matrix functions in Hermitian Clifford analysis. Some results of  and  are generalized in our paper.
Let be an 2n-dimensional ( ) real linear space with basis , be the -dimensional real linear space with basis
where N stands for the set and let denote the family of all order-preserving subsets of N in the above way. Now denote by and by for . The product on is defined by
where is the cardinal number of the set A, the number , , the symmetric difference set is also order-preserving in the above way, and is the coefficient of the -component of the Clifford number λ. Also, denote by . It follows at once from the multiplication rule (2.1) that is the identity element written now as 1 and, in particular,
Thus is a real linear, associative, but non-commutative algebra and it is called the Clifford algebra over . An involution is defined by
From (2.1) and (2.3), we have
The Euclidean space is embedded in by identifying with the Clifford vector given by
Note that the square of is scalar valued and equals the norm squared up to a minus sign: . The dual of is the vector-valued first order differential operator
called a Dirac operator. It is precisely this Dirac operator which underlies the notion of monogenicity of a function, a notion which is the higher dimensional counterpart of holomorphy in the complex plane. A function f defined and differentiable in an open region Ω of and taking values in is called (left) monogenic in Ω if . As the Dirac operator factorizes the Laplacian, , monogenicity can be regarded as a refinement of harmonicity. We refer to this setting as the orthogonal case, since the fundamental group leaving the Dirac operator invariant is the special orthogonal group , which is doubly covered by the group of the Clifford algebra . For this reason, the Dirac operator is also called rotation invariant. When allowing for complex constants, the set of generators produces the complex Clifford algebra , being the complexification of the real Clifford algebra , i.e. . Any complex Clifford number may be written as , , an observation leading to the definition of the Hermitian conjugation , where the bar notation stands for the usual Clifford conjugation in , i.e. the main anti-involution for which , . This Hermitian conjugation also leads to a Hermitian inner product and its associated norm on is given by and .
The above will be the framework for the so-called Hermitian Clifford analysis, yet a refinement of orthogonal Clifford analysis. An elegant way for introducing this setting consists in considering a so-called complex structure, i.e. a specific -element J for which (see [15-17]). Here, J is chosen to act upon the generators of the Clifford algebra as
With J one may associate two projection operators which will produce the main protagonists of the Hermitian setting by acting upon the corresponding objects in the orthogonal framework. First of all, the so-called Witt basis elements for the complex Clifford algebra are obtained through the action of on the orthogonal basis elements :
These Witt basis elements satisfy the Grassmann identities,
and the duality identities,
Next we identify a vector in with the Clifford vector and we denote by the action of the complex structure J on , i.e.
Note that the vectors and are orthogonal, the Clifford vectors and anti-commute. The actions of the projection operators on the Clifford vector then produce the Hermitian Clifford variables and its Hermitian conjugate :
which can be rewritten in terms of the Witt basis elements as
where n complex variables have been introduced, with complex conjugates , . Finally, the Hermitian Dirac operators and are derived from the orthogonal Dirac operator :
where we have introduced
In view of the Witt basis, the Hermitian Dirac operators are expressed as
involving the classical Cauchy-Riemann operators and their complex conjugates in the complex -planes, .
Finally observe that the Hermitian vector variables and Dirac operators are isotropic, since the Witt basis elements are, i.e.
whence the Laplacian allows for the decomposition
For further use, we introduce the Hermitian oriented surface elements and as follows:
where denotes the vector-valued oriented surface element and . They are explicitly given by means of the following differential forms of order :
and the corresponding oriented volume elements then read
We also consider the associated volume element , defined as
reflecting integration over the respective complex -planes, . One has
We still introduce the matrix
which will play the role of the differential form.
Definition 2.1 A continuously differentiable function f on an open region Ω of with values in is called a (left) h-monogenic function in Ω, iff it satisfies in Ω the system
or, equivalently, the system
The respective fundamental solutions of and are given by
where denotes the area of the unit sphere in . The transition from Hermitian Clifford analysis to a circulant matrix approach is essentially based on the following observation. Introducing the particular circulant matrices
where and . Then , where δ is the diagonal matrix with the Dirac delta distribution δ on the diagonal, may be considered as a fundamental solution of the matrix operator . This has also led to a theory of H-monogenic circulant matrix functions, the framework for this theory being as follows. Let , be continuously differentiable functions defined in Ω and taking values in , and consider the corresponding circulant matrix function
The ring of such matrix functions over is denoted by . In what follows, O will be denoting the matrix in with zero entries.
Definition 2.2 The matrix function is called (left) H-monogenic in Ω if and only if it satisfies in Ω the system .
The notions of continuity, differentiability, and integrability of have the usual component-wise meaning. In particular, we will need to defined in this way the classes , , of r times continuously differentiable functions over some suitable subset Ω of , stands for Hölder continuous circulant matrix functions over Ω. We introduce the non-negative function
where denotes the Clifford norm.
Definition 2.3 The matrix function ( ) is called (left) H-2-monogenic in Ω if and only if it satisfies in Ω the system .
In what follows we suppose
3 Some properties for H-monogenic circulant matrix functions
Theorem 3.1If the matrix functions isH-monogenic in Ω then
for each such that .
Proof Take such that . Apply Hermitian Cauchy’s integral formula I (in ). On the ball , we have
As , we apply the Hermitian Clifford-Stokes theorem (in ),
The result follows. □
The notions of continuity, differentiability, and integrability of have the usual component-wise meaning.
Theorem 3.2 (Liouville theorem)
If the matrix function isH-monogenic in and satisfies for all then must be a constant circulant matrix in .
Proof By Theorem 3.1, we have
where denotes the symmetric difference of and , is Lebesgue volume measure on , so that . The last expression above tends to 0 as . Thus and so is a constant circulant matrix. □
Theorem 3.3 (Maximum modulus theorem)
Let the matrix functions be aH-monogenic in the open and connected set Ω. If there exists a point such that
for all , then must be constant circulant matrix in Ω.
Proof Put and consider the subset of Ω given by
Since , then . So let ; this implies that . As is continuous in Ω, there exists an such that . This means that is relatively closed in Ω.
Now take and such that . By Theorem 3.1, we have
we then have
Applying Hölder’s inequality,
which yields for all , this means that and hence that is relatively open in Ω. As Ω is supposed to be connected it follows that .
Now if then clearly for all . For , since is H-monogenic in Ω, we have
then for all , and . Hence we obtain and for all . For all we have
and by (3.7), differentiating twice, we get
Summing up over yields
we have ( ) in Ω for all all . Thus , are constants in Ω. The result follows. □
Corollary 3.4Let Ω be a bounded open set in and suppose that , are functions in and isH-monogenic in Ω. Then
4 Higher order Hermitian Borel-Pompeiu formula in Hermitian Clifford analysis
Integral representation formulas in Clifford analysis have been well developed in [3,23-25], etc. These integral representation formulas are powerful tools. In this section, we get the explicit expression of the kernel function for and then get the explicit integral representation formulas for functions in Hermitian Clifford analysis. These explicit integral representation formulas play an important role in studying the further properties of the functions in Hermitian Clifford analysis.
In what follows, we denote
where denotes the area of the unit sphere in .
Lemma 4.1Let be as in (4.5). Then .
Proof The identity is obtained by straightforward calculation. □
Lemma 4.2Denote , , then
Lemma 4.3Let and be as in (4.4) and (4.3). Then
Proof In view of Lemma 4.2, the identity is obtained by straightforward calculation. □
Theorem 4.4 (Higher order Hermitian Borel-Pompeiu formula)
Suppose is a 2n-dimensional compact differentiable and oriented manifold with smooth boundary∂Γ, and are functions in and is the matrix function. It then follows that
Proof First let . It then follows from the Stokes formula, which can be found in , that we have
then the left-hand side of the stated formula apparently equals zero.
Now, let and take such that . Invoking the previous case, we may then write
Here we take the limits for . In view of the weak singularity of the third term of (4.11) yields
since the integrand only contains functions which are integrable on Γ. Furthermore we may write
Combining the Stokes formula in Hermitian Clifford analysis with
where is defined as in (4.2).
It is clear that
Then the result follows. □
Theorem 4.5If the matrix function isH-2-monogenic in Ω then
Proof Since is H-2-monogenic in Ω, in view of Theorem 4.4, the result follows. □
Theorem 4.6Let be an open ball centered at with radiusRin , and the matrix function isH-2-monogenic in , then for all
Theorem 4.7 (Mean value theorem for H-2-monogenic matrix function)
If the matrix function isH-2-monogenic in Ω then
for each such that .
Proof Take such that , by Theorem 4.5 we get
Combining with the Stokes formula in Hermitian Clifford analysis, is H-2-monogenic in Ω, Lemma 4.3 with , we have
The proof is done. □
Corollary 4.8If the matrix function isH-2-monogenic in and satisfies for all , then must be a constant circulant matrix in .
Proof The proof is similar to the method in Theorem 3.2. □
Suppose Ω is an open bounded non-empty subset of with a Liapunov boundary ∂Ω, we usually write and . The notations and will be reserved for Clifford vectors associated to points , while their Hermitian counterparts are denoted and . By means of the matrix approach sketched above, the following Hermitian Plemelj-Sokhotski formula.
We shall introduce the following matrix operators:
Let . Then the boundary values of the Hermitian Cauchy integral are given by
Theorem 4.10Let be an open ball centered at , with radiusRin , , in , , . Then in .
Proof We only need to prove that for any , . Define , . For any , taking constants , is a ball with the center at and radius δ such that . Obviously, is a Liapunov boundary. Using the Hermitian Borel-Pompeiu formula, we have
Using Lemma 4.9, for , we obtain
Combining (4.26) with (4.27), we get
Therefore , and the result follows. □
Theorem 4.11Let be an open ball centered at , with radiusRin , , in , and satisfies the following conditions:
where , then in .
Proof In view of the weak singularity of , combining Theorem 4.6 with Lemma 4.9, the theorem can be similarly proved similarly to Theorem 4.10. □
Theorem 4.12Let , in ,
where , ( ), then for
where be a constant circulant matrix.
5 Riemann boundary value problem for H-monogenic functions
An Riemann boundary value problem for H-monogenic functions is denoted as follows: