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 4.1 Let be as in (4.5). Then .
Proof The identity is obtained by straightforward calculation. □
Lemma 4.2 Denote , , then
Lemma 4.3 Let 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 4.5 If the matrix function is H-2-monogenic in Ω then
Proof Since is H-2-monogenic in Ω, in view of Theorem 4.4, the result follows. □
Theorem 4.6 Let be an open ball centered at with radius R in , and the matrix function is H-2-monogenic in , then for all
Theorem 4.7 (Mean value theorem for H-2-monogenic matrix function)
If the matrix function is H-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 4.8 If the matrix function is H-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:
Lemma 4.9 [18, 21]
Let . Then the boundary values of the Hermitian Cauchy integral are given by
Theorem 4.10 Let be an open ball centered at , with radius R in , , 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 4.11 Let be an open ball centered at , with radius R in , , 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 4.12 Let , in ,
where , (), then for
where be a constant circulant matrix.