Extension Theorem for Complex Clifford Algebras-Valued Functions on Fractal Domains

Monogenic extension theorem of complex Clifford algebras-valued functions over a bounded domain with fractal boundary is obtained. The paper is dealing with the class of Hölder continuous functions. Applications to holomorphic functions theory of several complex variables as well as to that of the so-called biregular functions will be deduced directly from the isotonic approach.

It is well known that methods of Clifford analysis, which is a successful generalization to higher dimension of the theory of holomorphic functions in the complex plane, are a powerful tool for study boundary value problems of mathematical physics over bounded domains with sufficiently smooth boundaries; see [1–3].

One of the most important parts of this development is the particular feature of the existence of a Cauchy type integral whose properties are similar to its famous complex prototype. However, if domains with boundaries of highly less smoothness (even nonrectifiable or fractal) are allowed, then customary definition of the Cauchy integral falls, but the boundary value problems keep their interest and applicability. A natural question arises as follows.

Can we describe the class of complex Clifford algebras-valued functions from Hölder continuous space extending monogenically from the fractal boundary of a domain through the whole domain?

In [4] for the quaternionic case and in [5–7] for general complex Clifford algebra valued functions some preliminaries results are given. However, in all these cases the condition ensures that extendability is given in terms of box dimension and Hölder exponent of the functions space considered.

In this paper we will show that there is a rich source of material on the roughness of the boundaries permitted for a positive answer of the question which has not yet been exploited, and indeed hardly touched.

At the end, applications to holomorphic functions theory of several complex variables as well as to the so-called biregular functions (to be defined later) will be deduced directly from the isotonic approach.

The above motivation of our work is of more or less theoretical mathematical nature but it is not difficult to give arguments based on an ample gamma of applications.

Indeed, the M. S. Zhdanov book cited in [8] is a translation from Russian and the original title means literally "The analogues of the Cauchy-type integral in the Theory of Geophysics Fields". In this book is considered, as the author writes, one of the most interesting questions of the Potential plane field theory, a possibility of an analytic extension of the field into the domain occupied by sources.

He gives representations of both a gravitational and a constant magnetic field as such analogues in order to solve now the spatial problems of the separation of field as well as analytic extension through the surface and into the domain with sources.

Our results can be applied to the study of the above problems in the more general context of domains with fractal boundaries, but the detailed discussion of this technical point is beyond the scope of this paper.

Let be an orthonormal basis of the Euclidean space .

The complex Clifford algebra, denoted by , is generated additively by elements of the form

where is such that , and so the complex dimension of is . For , is the identity element.

For , the conjugation and the main involution are defined, respectively, as

If we identify the vectors of with the real Clifford vectors , then may be considered as a subspace of .

The product of two Clifford vectors splits up into two parts:

where

Generally speaking, we will consider -valued functions on of the form

where are -valued functions. Notions of continuity and differentiability of are introduced by means of the corresponding notions for its complex components .

In particular, for bounded set , the class of continuous functions which satisfy the Hölder condition of order in will be denoted by .

Let us introduce the so-called Dirac operator given by

It is a first-order elliptic operator whose fundamental solution is given by

where is the area of the unit sphere in .

If is open in and , then is said to be monogenic if in . Denote by the set of all monogenic functions in . The best general reference here is [9].

We recall (see [10]) that a Whitney extension of , being compact in , is a compactly supported function such that and

Here and in the sequel, we will denote by certain generic positive constant not necessarily the same in different occurrences.

The following assumption will be needed through the paper. Let be a Jordan domain, that is, a bounded oriented connected open subset of whose boundary is a compact topological surface. By we denote the complement domain of .

By definition (see [11]) the box dimension of , denoted by , is equal to where stands for the least number of -balls needed to cover .

The limit above is unchanged if is thinking as the number of -cubes with intersecting . A cube is called a -cube if it is of the form: where are integers.

Fix , assuming that the improper integral converges. Note that this is in agreement with [12] for to be -summable.

Observe that a -summable surface has box dimension . Meanwhile, if has box dimension less than , then is -summable.

We begin this section with a basic result on the usual Cliffordian Théodoresco operator defined by

If such that , which we may assume, then it follows that and we may choose such that . If for such we can prove that then by in [3, Proposition ] it follows that represents a continuous function in . Moreover, for any , which is due to the fact that .

In the remainder of this section we assume that .

3.1. Monogenic Extension Theorem

Theorem 3.1.

If is the trace of , then

Conversely, assuming that (3.2) holds, then is the trace of , for any .

Proof.

Let and define

and .

Note that the boundary of , denoted by , is actually composed by certain faces (denoted by ) of some cubes . We will denote by , the outward pointing unit normal to and , respectively, in the sense introduced in [13].

Let and let be so large chosen that and for , where is a cube of . Here and below denotes the diameter of as a subset of .

Let , a cube containing , and such that .

Since , , we have

Let be an -dimensional face of and the -cube containing ; then if , we have

Each face of is one of those of some . Therefore, for

Since , we get

By Stokes formula we have

Therefore

The same conclusion can be drawn for . The only point now is to note that for .

Finally, due to the fact that

we prove that , and the second assertion follows directly by taking .

The finiteness of the last sum follows from the -summability of together with the fact that .

For the following analogous result can be obtained.

Theorem 3.2.

Let . If is the trace of and , then

Conversely, assuming that (3.11) holds, then is the trace of , for any .

3.2. Isotonic Extension Theorem

For our purpose we will assume that the dimension of the Euclidean space is even whence we will put from now on.

In a series of recent papers, so-called isotonic Clifford analysis has emerged as yet a refinement of the standard case but also has strong connections with the theory of holomorphic functions of several complex variables and biregular ones, even encompassing some of its results; see [14–18].

Put

then a primitive idempotent is given by

We have the following conversion relations:

with (complex Clifford algebra generated by ).

Note that for one also has that

Let us introduce the following real Clifford vectors and their corresponding Dirac operators:

The function is said to be isotonic in if and only if is continuously differentiable in and moreover satisfies the equation

We will denote by the set of all isotonic functions in .

We find ourselves forced to introduce two extra Cauchy kernels, defined by

Now we may introduce the isotonic Théodoresco transform of a function to be

It is straightforward to deduce that

Theorem 3.3.

If , is the trace of , then

Conversely, assuming that (3.21) holds, then is the trace of , for any .

Proof.

Let be an isotonic extension of to such that . Then, is a monogenic extension of to , which obviously belongs to . Therefore

by Theorem 3.1.

We thus get

the first equality being a direct consequence of (3.20). According to (3.15) we have (3.21), which is the desired conclusion.

On account of Theorem 3.1 again, the converse assertion follows directly by taking , and the proof is complete.

Remark 3.4.

Theorems 3.1 and 3.3 extend the results in [4–7], since the restriction putted there () implies that of this paper.

In this last section, we will briefly discuss two particular cases which arise when considering (3.17).

Case 1.

It is easily seen that if takes values in the space of scalars , then is isotonic if and only if

which means that is a holomorphic function with respect to the complex variables .

Case 2.

If , isotonic function, takes values in the real Clifford algebra , then

or, equivalently, by the action of the main involution on the second equation we arrive to the overdetermined system:

whose solutions are called biregular functions. For a detailed study we refer the reader to [19–21].

The proof of Theorem 3.3 may readily be adapted to establish analogous results for both holomorphic and biregular functions context. Clearly, we prove that if we replace by a -valued, respectively, -valued function, such that (3.21) holds, then there exists an isotonic extension , which, by using the classical Dirichlet problem, takes values precisely in or , respectively. On the other direction the proof is immediate. The corresponding statements are left to the reader.

The topic covered here has been initiated while the first two authors were visiting IMPA, Rio de Janeiro, in July of 2009. Ricardo Abreu and Juan Bory wish to thank CNPq for financial support. Ricardo Abreu wishes to thank the Faculty of Ingeneering, Universidad Diego Portales, Santiago de Chile, for the kind hospitality during the period in which the final version of the paper was eventually completed. This work has been partially supported by CONICYT (Chile) under FONDECYT Grant 1090063.

1. Gürlebeck, K, Sprössig, W: Quaternionic Analysis and Elliptic Boundary Value Problems, International Series of Numerical Mathematics,p. 253. Birkhäuser, Basel, Switzerland (1990)

2. Gürlebeck, K, Sprössig, W: Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley & Sons, New York, NY, USA (1997)

3. Gürlebeck, K, Habetha, K, Sprössig, W: Holomorphic Functions in the Plane and n-Dimensional Space, Birkhäuser, Basel, Switzerland (2007)

4. Abreu-Blaya, A, Bory-Reyes, J, Moreno-García, T: Holomorphic extension theorems in Lipschitz domains of . Advances in Applied Clifford Algebras. 20(1), 1–12 (2010). Publisher Full Text

5. Abreu-Blaya, A, Bory-Reyes, J, Peña, DP, Sommen, F: Biregular extendability via isotonic Clifford analysis. Mathematical Methods in the Applied Sciences. 33(4), 384–393 (2010)

6. Abreu-Blaya, R, Bory-Reyes, J, Moreno-García, T: Teodorescu transform decomposition of multivector fields on fractal hypersurfaces. Wavelets, Multiscale Systems and Hypercomplex Analysis, Operator Theory: Advances and Applications, pp. 1–16. Birkhäuser, Basel, Switzerland (2006).

7. Abreu-Blaya, R, Bory-Reyes, J, Moreno-García, T: A biregular extension theorem from a fractal surface in . to appear in Georgian Mathematical Journal

8. Shapiro, M, Vasilievski, NL, Zhdanov, MS: Space analogs of the Cauchy type integrals and the quaternion theory. In: Zhdanov MS (ed.) Integral Transforms in Geophysics,p. 367. Springer, Heidelberg, Germany (1988) (Appendix A: 344–350)

9. Brackx, F, Delanghe, R, Sommen, F: Clifford Analysis, Research Notes in Mathematics,p. x+308. Pitman, Boston, Mass, USA (1982)

10. Stein, EM: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30,p. xiv+290. Princeton University Press, Princeton, NJ, USA (1970)

11. Falconer, KJ: The Geometry of Fractal Sets, Cambridge Tracts in Mathematics,p. xiv+162. Cambridge University Press, Cambridge, UK (1986)

12. Harrison, J, Norton, A: The Gauss-Green theorem for fractal boundaries. Duke Mathematical Journal. 67(3), 575–588 (1992). Publisher Full Text

13. Federer, H: Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften,p. xiv+676. Springer, New York, NY, USA (1969)

14. Abreu-Blaya, R, Bory-Reyes, J: A Martinelli-Bochner formula on fractal domains. Archiv der Mathematik. 92(4), 335–343 (2009). Publisher Full Text

15. Abreu-Blaya, R, Bory-Reyes, J, Peña Peña, D, Sommen, F: The isotonic Cauchy transform. Advances in Applied Clifford Algebras. 17(2), 145–152 (2007). Publisher Full Text

16. Abreu-Blaya, A, Bory-Reyes, J, Peña, DP, Sommen, F: Holomorphic extension theorems in Lipschitz domains of . Advances in Applied Clifford Algebras. 20(1), 1–12 (2010). Publisher Full Text

17. Bory-Reyes, J, Peña Peña, D, Sommen, F: A Davydov theorem for the isotonic Cauchy transform. Journal of Analysis and Applications. 5(2), 109–121 (2007)

18. Sommen, F, Peña Peña, D: Martinelli-Bochner formula using Clifford analysis. Archiv der Mathematik. 88(4), 358–363 (2007). Publisher Full Text

19. Brackx, F, Pincket, W: A Bochner-Martinelli formula for the biregular functions of Clifford analysis. Complex Variables: Theory & Application. 4(1), 39–48 (1984). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

20. Brackx, F, Pincket, W: The biregular functions of Clifford analysis: some special topics. Clifford Algebras and Their Applications in Mathematical Physics (Canterbury, 1985), NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, pp. 159–166. Reidel, Dordrecht, The Netherlands (1986)

21. Brackx, F, Pincket, W: Two Hartogs theorems for nullsolutions of overdetermined systems in Euclidean space. Complex Variables: Theory & Application. 4(3), 205–222 (1985). PubMed Abstract | Publisher Full Text | PubMed Central Full Text