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 higherdimensional function theory centered around the simultaneous null solutions of two firstorder vectorvalued 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 AhlforsDavid regular boundary were considered.
1. Introduction
Clifford analysis is a higherdimensional 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 vectorvalued 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 [6] 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 [16] 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 [12]), 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 [23] 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. Preliminaries
2.1. Some Elements of Hermitean Clifford Analysis
Let be an orthonormal basis of Euclidean space and consider the complex Clifford algebra constructed over . The noncommutative or geometric multiplication in is governed by the following rules:
The Clifford algebra thus is generated additively by elements of the form , where is such that , while for , one puts , the identity element. The dimension of thus is . Any Clifford number may thus be written as , , and its Hermitean conjugate is defined by
where the bar denotes the real Clifford algebra conjugation, that is, the main antiinvolution 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 vectorvalued firstorder Dirac operator , factorizing the Laplacian: ; it is precisely this Dirac operator which underlies the notion of monogenicity of a function, the higherdimensional 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 algebravalued function , defined and differentiable in an open region of , is then called (left) monogenic in if and only if in .
The transition from Euclidean Clifford analysis as described above to the Hermitean Clifford setting is essentially based on the introduction of a complex structure . This is a particular element, satisfying . Since such an element cannot exist when the dimension of the vector space is odd, we will put from now on. In terms of the chosen orthonormal basis, a particular realization of the complex structure may be and , . Two projection operators associated to this complex structure then produce the main objects of Hermitean Clifford analysis by acting upon the corresponding objects in the Euclidean setting; see [11, 12]. First of all, the vector space thus decomposes as into two isotropic subspaces. The real Clifford vector is now denoted by
and its corresponding Dirac operator by
while we will also consider their socalled "twisted" counterparts, obtained through the action of , that is,
As was the case with , a notion of monogenicity may be associated in a natural way to as well. The projections of the vector variable and the Dirac operator on the spaces then give rise to the Hermitean Clifford variables and , given by
and (up to a factor) to the Hermitean Dirac operators and given by
(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]).
Definition 2.1.
A continuously differentiable function in with values in is called left Hermitean monogenic (or left monogenic) in , if and only if it satisfies in the system
or, equivalently, the system
In a similar way right monogenicity is defined. Functions which are both left and right monogenic are called twosided monogenic. This definition inspires the statement that monogenicity constitutes a refinement of monogenicity, sincemonogenic functions (either left or right) are monogenic w.r.t. both Dirac operators and .
In what follows, we will systematically take to be a socalled 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
The fundamental solutions of the Dirac operators and are, respectively, given by
where denotes the surface area of the unit sphere in . Introducing the functions and , explicitly given by
it is directly seen that and are not the fundamental solutions to the respective Hermitean Dirac operators and . Surprisingly, however, introducing the particular circulant matrices
where is the Dirac delta distribution, one obtains that , so that may be considered as a fundamental solution of in a matricial context, see, for example, [8, 16, 18]. Moreover, the Dirac matrix in some sense factorizes the Laplacian, since
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.
Definition 2.2.
Let be continuously differentiable functions defined in and taking values in , and consider the matrix function:
Then is called left (resp., right) monogenic in if and only if it satisfies in the system
Here denotes the matrix with zero entries.
Explicitly, the system for left monogenicity reads:
Again, a notion of twosided 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
Let be an arbitrary subset of . Then for any its Hausdorff measure may be defined by
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].
Frequently, however, see, for example, [27], the socalled box dimension is more appropriated than the Hausdorff dimension to measure the roughness of a given set . By definition, the box dimension of a compact set is equal to
where stands for the minimal number of balls needed to cover . Note that the limit in (2.18) remains unchanged if is replaced by the number of cubes, with , intersecting . For completeness we recall that a cube is called a cube if it is of the form
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 socalled rectifiable sets (see [28]), 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
From now on we reserve the notations and for Clifford vectors associated to points in . We may then consider the Euclidean Téodorescu transforms and of a function , assumed to be integrable in , given by
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 wellknown properties established in the complex plane.
To this end, let be a valued function defined on , which satisfies a Hölder condition of order , that is, , where , and denote by the socalled Whitney extension of from to the whole of (see [29]). We recall that the Whitney extension of is a compactly supported function for which it holds that and
We then first formulate an auxiliary result.
Lemma 3.1.
and are integrable in for any .
Proof.
We only give the main lines of the proof; for details we refer the reader to [22, Lemma ]. In the notation of [30], let be the Whitney partition of by means of cubes. We then have
On the other hand, (3.2) implies that for
since . Now, invoking the fact that the number of cubes appearing in is less than , while by definition of ,
for any , we arrive at
the last series being convergent for . In view of the arbitrary choice of , this concludes the proof.
Now, take such that
and then it holds that ; whence, on account of the previous lemma, there exist exponents such that and are integrable in . From this observation it then follows that, for , with as in (3.7), both and belong to , for any satisfying
due to the fact that the Téodorescu transform maps the space of integrable functions with compact support to if (see, e.g., [5]). The following result then holds.
Proposition 3.2.
For , with as in (3.7), consider
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 .
Proof.
For simplicity we restrict ourselves to , the proof for running along similar lines. The continuity of on the closed domains follows from the fact that for any satisfying (3.8). On the other hand, a direct calculation shows that and that
where
Finally, the monogenicity of is a direct consequence of the wellknown fact that the Téodorescu transform constitutes a right inverse of the Dirac operator.
Summarizing, any function , with as in (3.7), can be decomposed as
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 [22].
Theorem 3.3.
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.
Theorem 3.4.
Suppose that
Then, for any in , , there exists a unique decomposition (3.12) of class .
Proof.
The existence being shown above, it remains to prove the uniqueness. To this end, assume that admits two decompositions of class , denoted by and , respectively. Then
implying that
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
A first step in the solution of the Hermitean matrix decomposition problem is the introduction of the matricial Hermitean Téodorescu transform:
where is the associated volume element given by
and denote the Hermitean counterparts of , that is,
It clearly holds that .
A direct calculation reveals that the Hermitean Téodorescu transform can be expresed in terms of the Euclidean Téodorescu transforms and as follows (see [15]):
In particular, for the special case of the matrix function (i.e., and ) this expression reduces to
In what follows we will denote by the Whitney extension of , that is,
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.
Theorem 4.1.
If , with , then .
If , then
If with as in (3.7), then
for any satisfying (3.8).
5. The HMonogenic Decomposition Problem
We are now in the possibility to treat the Hmonogenic decomposition problem; it means to study under which conditions a given matrix function can be decomposed as
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.
Theorem 5.1.
Let , with as in (3.7). Then admits the Hermitean monogenic decomposition (5.1), where the components are explicitly given by
Proof.
On account of the assumption on , it follows that belongs to , for and , simultaneously. Then, the Hölder continuity of directly follows from Theorem 4.1, (i) and (iii). Next, the matrix inversion formula (ii) in Theorem 4.1 yields
showing the monogenicity of in , respectively.
In order to rephrase Theorem 5.1 in the monogenic setting, as studied in [23], we only need to ensure that
or, equivalently, that
which, by means of some direct calculations, can be rewritten as
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.
Theorem 5.2.
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.
Remark 5.3.
Even though the decomposition (5.1) is not unique in general, it will be so in the corresponding class:
for ; see Theorem 3.4.
Remark 5.4.
When is sufficiently regular, for example, AhlforsDavid regular, the expressions (5.2) reduce to the ones obtained in [17] in terms of the matricial Hermitean Cauchy integral, the latter being easily obtained using the Hermitean BorelPompeiu formula, as proved in [16]. Indeed, applying this BorelPompeiu formula to , we obtain
where is the Hermitean Cauchy integral given by
Here, the additional circulant matrix
contains (up to a factor) the Hermitean projections and of the outward unit normal vector at the point , while the matrix Hausdorff measure is given by
Since , we have
or, using the notations of (5.2),
which coincides with [17, equation ()].
Acknowledgments
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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