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.
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,
From (2.1) and (2.3), we have
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,
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 we have introduced
In view of the Witt basis, the Hermitian Dirac operators are expressed as
Finally observe that the Hermitian vector variables and Dirac operators are isotropic, since the Witt basis elements are, i.e.
and the corresponding oriented volume elements then read
We still introduce the matrix
which will play the role of the differential form.
or, equivalently, the system
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 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
In what follows we suppose
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. □
Theorem 3.2 (Liouville theorem)
Proof By Theorem 3.1, we have
Theorem 3.3 (Maximum modulus theorem)
we then have
Applying Hölder’s inequality,
and by (3.7), differentiating twice, we get
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
Proof The identity is obtained by straightforward calculation. □
Proof In view of Lemma 4.2, the identity is obtained by straightforward calculation. □
Theorem 4.4 (Higher order Hermitian Borel-Pompeiu formula)
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.
since the integrand only contains functions which are integrable on Γ. Furthermore we may write
Combining the Stokes formula in Hermitian Clifford analysis with
It is clear that
Then the result follows. □
Theorem 4.7 (Mean value theorem for H-2-monogenic matrix function)
The proof is done. □
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:
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
Combining (4.26) with (4.27), we get
5 Riemann boundary value problem for H-monogenic functions
Theorem 5.1The Riemann boundary value problem (5.1) is solvable and the solution can be written as
Furthermore, we denote
can be changed into
and if we denote
From (5.5) and (5.7) we have
On the other hand, it can be directly proved that (5.2) is the solution of (5.1), and the proof is done. □
6 Riemann boundary value problem for H-2-monogenic function in Hermitian Clifford analysis
Theorem 6.1The Riemann boundary value problem (6.1) is solvable and the solution is given by
Combining (6.6) with (6.7) we then get
then we obtain
Combining (6.7) with (6.10) we arrive at the proposed result.
On the other hand, it can be directly proved that (6.2) are the solution of (6.1) and the proof is done. □
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
This paper is supported by National Natural Science Foundation of China (11271175), the AMEP, and DYSP of Linyi University. The authors would like to thank the referees for their valuable suggestion and comments.
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