Hermitean Cauchy Integral Decomposition of Continuous Functions on Hypersurfaces
© Ricardo Abreu Blaya et al. 2008
Received: 12 August 2008
Accepted: 23 October 2008
Published: 16 November 2008
We consider Hölder continuous circulant matrix functions defined on the Ahlfors-David regular boundary of a domain in . The main goal is to study under which conditions such a function can be decomposed as , where the components are extendable to two-sided -monogenic functions in the interior and the exterior of , respectively. -monogenicity is a concept from the framework of Hermitean Clifford analysis, a higher dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. -monogenic functions then are the null solutions of a matrix Dirac operator, having these Hermitean Dirac operators as its entries; such functions have been crucial for the development of function theoretic results in the Hermitean Clifford context.
Clifford analysis essentially is a higher dimensional function theory offering both a generalization of the theory of holomorphic functions in the complex plane and a refinement of classical multidimensional harmonic analysis. The standard case, nowadays also called Euclidean Clifford analysis, focuses on monogenic functions, that is, the null solutions of the vector-valued Dirac operator , factorizing the -dimensional Laplacian: . Here is an orthonormal basis for the quadratic space underlying the construction of the real Clifford algebra . The fundamental group leaving the Dirac operator invariant is the special orthogonal group , doubly covered by the spin( ) group of the Clifford algebra . For this reason, the Dirac operator is called a rotation invariant operator. Standard references for Euclidean Clifford analysis are [1–4].
In a series of recent papers, the so-called Hermitean Clifford analysis has emerged as yet a refinement of the Euclidean case. One of the ways for introducing it is by considering the complex Clifford algebra and a so-called complex structure on it, that is, an -element for which . It is precisely the requirement that such a complex structure exists, which forces the dimension of the underlying vector space to be even: . The resulting function theory focuses on the simultaneous null solutions of two complex Hermitean Dirac operators and which no longer factorize, but still decompose the Laplace operator in the sense that . The fundamental group symmetry of this system breaks down to the action of the special unitary group. The study of complex Dirac operators was initiated in [5–8]; a systematic development of the associated function theory, including the invariance properties with respect to the underlying Lie groups and Lie algebras, is still in full progress (see, e.g., [9–13]).
In the paper , a Cauchy integral formula for Hermitean monogenic functions was established, obviously an essential result in the function theory. However, as in some very particular cases Hermitean monogenicity turns out to be equivalent with anti-holomorphy in complex variables (see ), it was predictable that such a representation formula could not, in the present setting, take the traditional form as in the complex plane or in Euclidean Clifford analysis. Indeed, a matrix approach had to be followed in order to obtain the desired result, leading to the concept of (left or right) -monogenic functions, introduced as circulant matrix functions, which are (left or right) null solutions of a circulant matrix Dirac operator, having the Hermitean Dirac operators and as its entries.
Although the -monogenic system thus arose as an auxiliary concept in Hermitean Clifford analysis, it deserves to be further studied for its own intrinsic value. In this paper, we consider Hölder continuous circulant matrix functions defined on the Ahlfors-David regular boundary of a domain in , and we investigate under which conditions such a function can be decomposed as , where the components are extendable to two-sided -monogenic functions in the interior and the exterior of , respectively. Such type of decomposition (or "jump") problem was considered already in Euclidean Clifford analysis in, for example, .
The present decomposition problem is discussed using the matrix Cauchy integral of (see also ) and its singular version , called the Hilbert transform; they are shown to be related to each other by Plemelj-Sokhotzki type formulae for the continuous boundary values of . Moreover, a result is obtained connecting the two-sided -monogenicity of a function in to a conservation law for the Hilbert transforms and of its trace on the boundary.
where the bar denotes the usual real Clifford algebra conjugation, that is, the main anti-involution for which , and denotes the standard complex conjugation. Note that, as any complex Clifford number may also be written as , , the Hermitean conjugation also takes the form .
where denotes the unit sphere containing vectors for which . The group yields a double cover for the orthogonal group , defined by the map , with . For a -valued function , the induced action of a spin element is given by ; it is well known (see ) that the Dirac operator commutes with this action, whence we call it rotation invariant.
The transition from the Euclidean Clifford setting described above to Hermitean Clifford analysis now is essentially based on the introduction of a complex structure . This is a particular element of , satisfying . Note that such an element cannot exist when the dimension of the underlying vector space is odd, whence from now on, we will put . In terms of our basis, a particular realization of the complex structure is given by and . The two projection operators associated to this complex structure then produce the main objects in the Hermitean Clifford setting by acting upon the corresponding objects in the Euclidean one. First of all, the vector space thus decomposes as into two isotropic subspaces.
In a similar way, right h-monogenicity is defined. Functions which are both left and right h-monogenic are called two-sided h-monogenic.
This definition inspires the statement that h-monogenicity constitutes a refinement of monogenicity.
its definition involving the spin element corresponding to the complex structure: . It has been proved that this group constitutes a realization in the Clifford algebra of the unitary group , and moreover, that the Hermitean Dirac operators commute with its associated action. Less precisely, one thus says that these operators are invariant under the action of the unitary group, and so is the notion of h-monogenicity.
3. A Pair of Cauchy Integrals and Hilbert Transforms in the Euclidean Setting
For the sake of completeness, we recall some basic properties of the singular Cauchy integrals and , which are generalizations to the case of Clifford analysis of the properties established in the complex plane as follows:
Formulae (3.7) express the boundary values of the Cauchy integrals in terms of their singular versions. The study of this boundary behaviour, in the Euclidean Clifford analysis context, has been the subject of intensive research in the last years, see for example [19–23]. We must remark that these formulae also hold for a wider class of rectifiable surfaces, containing as proper subclasses for instance differentiable, chord-arc, piecewise smooth, Liapunov, and Lipschitz surfaces as well as simple Lipschitz graphs.
4. The Hermitean Cauchy Integral: A Matrix Approach
where is the Dirac delta distribution, one obtains that , so that may be considered as a fundamental solution of the operator in a matricial context. It was exactly this simple observation which has lead to the idea of following a matrix approach in order to establish a Cauchy integral formula and the related function theoretic results in the Hermitean Clifford setting, see [14, 16]. Moreover, it inspired the following definition.
In general, the -monogenicity of the matrix function does not imply the h-monogenicity of its entry functions and . However, choosing in particular and , the -monogenicity of the corresponding diagonal matrix is seen to be equivalent to the h-monogenicity of the function .
The above matrix approach will form the key towards the construction of a boundary value theory of h-monogenic functions. In what follows, we will restrict to left monogenicity, unless explicitly stated.
It is clear that, in general, will not turn out to be a diagonal matrix, whence its entries will not be h-monogenic functions. The particular situation where , giving rise to an interpretation in terms of h-monogenicity, is explicitly treated below.
Now consider the special case of the matrix function , or equivalently, of a single nonzero entry . The above decomposition problem (5.1) then obviously is strongly connected to the analogous problem for h-monogenic functions, as studied in . In order to be able to rephrase the obtained result (5.2) in the h-monogenic setting, we only need to ensure that is interpretable as an h-monogenic function. In view of this observation and of [24, Remark 1, Theorem 2.2] may be reformulated into the present setting as follows.
The next result deals with a necessary and sufficient condition for the extendability of a given matrix function on the hypersurface to an -monogenic function in or in , vanishing at infinity. It is clear that the answer will have to involve the projection operators and .
Now let and let . Then , while, according to Theorem 4.3, . Thus, , yielding (5.4). Conversely, assume that (5.4) holds and consider given by (5.2), for . Then is an -monogenic function in and again by Theorem 4.3, we have that , which proves (i). Similar considerations apply to (ii).
For the special case of the matrix function , condition (5.7) can be rephrased in terms of the entry function as , which exactly is the criterion for the existence of an -monogenic extension of to , obtained in . Similarly, in , (5.8) yields .
This section is devoted to the proof of a remarkable result, establishing a connection between two-sided -monogenicity of a function in a domain and the singular Cauchy integrals and of its trace on the boundary of . We still mention that, in the context of Euclidean Clifford analysis, a similar "conservation law" was obtained in  for two-sided monogenicity.
They are left and right -monogenic, respectively, and hence both are harmonic in . Combining Theorems 4.3 and 4.4, we can assert that and are also continuous on . As is harmonic in and , it follows that for . The proof is completed by showing that in .
7. Main Theorem
Assuming (i) to hold, we may directly check, invoking Theorem 5.2, that . We thus get that , which yields (ii) in view of Theorem 6.1. This completes the proof since the inverse implication is trivial.
This paper was written while R. Abreu Blaya and J. Bory Reyes were visiting the Department of Mathematical Analysis of Ghent University. They were supported by the Special Research Fund no. 01T00807, obtained for the collaboration between the Clifford Research Group Ghent and the Cuban Research Group in Clifford analysis, on a project entitled Boundary Values Theory in Clifford Analysis. The authors wish to thank all members of this Department for their kind hospitality.
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