Hermitean Téodorescu Transform Decomposition of Continuous Matrix Functions on Fractal Hypersurfaces
© Ricardo Abreu-Blaya et al. 2010
Received: 18 December 2009
Accepted: 12 April 2010
Published: 20 May 2010
We consider Hölder continuous circulant ( ) matrix functions defined on the fractal 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 -monogenic functions in the interior and the exterior of , respectively. -monogenicity are 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 matrix functions play an important role in the function theoretic development of Hermitean Clifford analysis. In the present paper a matricial Hermitean Téodorescu transform is the key to solve the problem under consideration. The obtained results are then shown to include the ones where domains with an Ahlfors-David regular boundary were considered.
Clifford analysis is a higher-dimensional function theory offering a generalization of the theory of holomorphic functions in the complex plane and, at the same time, a refinement of classical harmonic analysis. The standard case, also referred to as Euclidean Clifford analysis, focuses on the null solutions, called monogenic functions, of the vector-valued Dirac operator , which factorizes the -dimensional Laplacian: . Here is an orthonormal basis for the quadratic space underlying the construction of the real Clifford algebra , where the considered functions take their values. Since the Dirac operator is invariant with respect to the action of the orthogonal group , doubly covered by the Pin( ) group of the Clifford algebra , the resulting function theory is said to be rotation invariant. Standard references for Euclidean Clifford analysis are [1–5].
More recently, 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 , equipped with a complex structure, that is, an element for which . In fact, it is precisely in order to ensure that such a complex structure exists that the dimension of the underlying vector space is taken 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 . Since the system indeed constitutes a refinement of the original Euclidean Dirac equation, the fundamental group invariance of this system breaks down to a smaller group; it was shown in  that it concerns the unitary group . The study of complex Dirac operators was initiated in [7–10]; a systematic development of the associated function theory still is in full progress; see, for example, [6, 11–15].
In  a Cauchy integral formula for Hermitean monogenic functions was established, obviously an essential result in the development of the function theory. However, as in some very particular cases Hermitean monogenicity is equivalent with (anti)holomorphy in complex variables (see ), such a representation formula could not take the traditional form as in the complex plane or in Euclidean Clifford analysis. The matrix approach needed to obtain the desired result leads 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 was meanwhile also further studied itself; see also [15, 17, 18].
In the present paper, we consider Hölder continuous circulant matrix functions defined on the fractal boundary of a domain in , and we investigate under which conditions such a function can be decomposed as , where the components are extendable to -monogenic functions in the interior and the exterior of , respectively. This type of decomposition (or "jump") problem has already been considered in Euclidean Clifford analysis in, for example, [19–22] for domains with boundaries showing minimal smoothness, including some results for fractal boundaries as well. In  a similar decomposition problem for domains with fractal boundaries was considered in the Hermitean Clifford context, the approach, however, not being suited for a treatment of the circulant matrix situation. It turns out that the introduction of a matricial Hermitean Téodorescu transform is crucial to solve this problem.
2.1. Some Elements of Hermitean Clifford Analysis
where the bar denotes the real Clifford algebra conjugation, that is, the main anti-involution for which , and stands for the complex conjugate of the complex number .
Euclidean space is embedded in the Clifford algebra by identifying with the real Clifford vector given by , for which . The Fischer dual of the vector is the vector-valued first-order Dirac operator , factorizing the Laplacian: ; it is precisely this Dirac operator which underlies the notion of monogenicity of a function, the higher-dimensional counterpart of holomorphy in the complex plane. The functions under consideration are defined on (open subsets of) and take values in the Clifford algebra . They are of the form , where the functions are complex valued. Whenever a property such as continuity and differentiability is ascribed to , it is meant that all the components possess the cited property. A Clifford algebra-valued function , defined and differentiable in an open region of , is then called (left) monogenic in if and only if in .
(see [6, 11]). Observe for further use that the Hermitean vector variables and Dirac operators are isotropic, that is, and , whence the Laplacian allows for the decomposition , while also . These objects lie at the core of the Hermitean function theory by means of the following definition (see, e.g., [6, 11]).
In a similar way right -monogenicity is defined. Functions which are both left and right -monogenic are called two-sided -monogenic. This definition inspires the statement that -monogenicity constitutes a refinement of monogenicity, since -monogenic functions (either left or right) are monogenic w.r.t. both Dirac operators and .
In what follows, we will systematically take to be a so-called Jordan domain, that is, a bounded oriented connected open subset of , the boundary of which is a compact topological surface. Note that, in the case , this notion coincides with the usual one of a Jordan domain in the complex plane. For further use, we also introduce the notation , and .
2.2. Some Elements of the Matricial Hermitean Clifford Setting
where is the usual Laplace operator in . It was exactly this simple observation which leads to the idea of following a matrix approach in order to establish integral representation formulae in the Hermitean setting; see [15, 16]. Moreover, it inspired the following definition.
Here denotes the matrix with zero entries.
Again, a notion of two-sided -monogenicity may be defined similarly. However, unless mentioned explicitly, we will only work with left -monogenic matrix functions. This matrix approach has also been successfully applied in [17, 24] for the construction of a boundary values theory of -monogenic functions.
Observe however that the -monogenicity of the matrix function does not imply the -monogenicity of its entry functions and . Nevertheless, choosing in particular and , the -monogenicity of the corresponding diagonal matrix, denoted by , is seen to be equivalent to the -monogenicity of the function . Moreover, considering the matricial Laplacian introduced in (2.13), one may call a matrix function harmonic if and only if it satisfies the equation . Each -monogenic matrix function then is harmonic, ensuring that its entries are harmonic functions in the usual sense.
In general, notions of continuity, differentiability, and integrability of are introduced by means of the corresponding notions for its entries. In what follows, we will in particular use the notations , , and for the class of Hölder continuous and -integrable circulant matrix functions, respectively.
2.3. Some Elements of Fractal Geometry
where the infimum is taken over all countable -coverings of with open or closed balls. Note that, for , the Hausdorff measure coincides, up to a positive multiplicative constant, with the Lebesgue measure in .
Now, let be a compact subset of . The Hausdorff dimension of , denoted by , is then defined as the infimum of all such that . For more details concerning the Hausdorff measure and dimension we refer the reader to [25, 26].
where and are integers. The box dimension and the Hausdorff dimension of a given compact set can be equal, which is, for instance, the case for the so-called -rectifiable sets (see ), but this is not the case in general, where we have that .
In what follows we will assume the boundary of our Jordan domain to have Hausdorff dimension . This includes the case when is fractal in the sense of Mandelbrot, that is, when .
3. A Pair of Euclidean Téodorescu Transforms
where and are oriented volume elements on , for which it is easily checked that . For the sake of completeness, we recall some basic properties of and , which are generalizations to the case of Clifford analysis of the well-known properties established in the complex plane.
We then first formulate an auxiliary result.
and are -integrable in for any .
the last series being convergent for . In view of the arbitrary choice of , this concludes the proof.
due to the fact that the Téodorescu transform maps the space of -integrable functions with compact support to if (see, e.g., ). The following result then holds.
With denoting the characteristic function of the set . Then and are monogenic in and in , with respect to and , respectively. They are continuous in the corresponding closed domains, vanish at infinity, and show jump over the boundary .
Finally, the monogenicity of is a direct consequence of the well-known fact that the Téodorescu transform constitutes a right inverse of the Dirac operator.
where the components are extendable to monogenic functions in the interior and the exterior of the domain , with respect to and , respectively. Note that a decomposition of type (3.12) is said to be of class if . The remaining question is whether the decomposition (3.12) is unique. In order to investigate this, we will need the following version of the Dolzhenko theorem, as proved in .
Let the compact set be such that . Then, a function which is monogenic in is monogenic in the whole of .
We then arrive at the following result.
Then, for any in , , there exists a unique decomposition (3.12) of class .
Consequently, the function is monogenic in and belongs to , whence it is monogenic in on account of Theorem 3.3, while it also vanishes at . By Liouville's Theorem we conclude that .
4. A Matricial Hermitean Téodorescu Transform
It clearly holds that .
The following theorem then contains some of its basic properties of the matricial Hermitean Téodorescu transform. They can be proven using standard techniques applied to the present matrix context.
If , with , then .
for any satisfying (3.8).
5. The H-Monogenic Decomposition Problem
where the components , are Hölder continuous matrix functions on , which are Hermitean monogenically extendable to respectively, and moreover . The following theorem provides an answer to that question.
showing the -monogenicity of in , respectively.
This is precisely the condition under which a function has been found to admit an -monogenic decomposition (3.12); see [23, Theorem ]. This result may be reformulated into the present setting as follows.
Let , with as in (3.7), and consider the corresponding matrix function . Then admits the decomposition (5.1) in terms of -monogenic functions if and only if (5.4) holds.
for ; see Theorem 3.4.
which coincides with [17, equation ( )].
This paper was written during a scientific stay of the first author at the Clifford Research Group of the Department of Mathematical Analysis of Ghent University, supported by a "Visiting Postdoctoral Fellowship" of the Flemish Research Foundation. He wishes to thank the members of the Clifford Research Group for their kind hospitality during this stay.
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