Clifford analysis

In this paper we consider Jordan domains in real Euclidean spaces of higher dimension which have fractal boundaries. The case of decomposing a Hölder continuous multivector field on the boundary of such domains is obtained in closed form as sum of two Hölder continuous multivector fields harmonically extendable to the domain and to the complement of its closure respectively. The problem is studied making use of the Teodorescu transform and suitable extension of the multivector fields. Finally we establish equivalent condition on a Hölder continuous multivector field on the boundary to be the trace of a harmonic Hölder continuous multivector field on the domain.

We introduce mollifiers in Clifford analysis setting and construct a sequence of $\C^{\infinity}$-functions that approximate a $\gamma$-regular function and a solution to a non homogeneous BVP of an in homogeneous Dirac like operator in certain Sobolev spaces over bounded domains whose boundary is not that wild. One can extend the smooth functions upto the boundary if the domain has a $\C^1$-boundary and this is the case in the paper as we consider a domain whose boundary is a $\C^2$-hyper surface.

Conformal mappings in the plane are closely linked with holomorphic functions and their property of complex differentiability. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M ¨ obius transformations which are not mono- genic and therefore also not hypercomplex differentiable. But due to the equivalence between being hypercomplex differentiable and being

In this paper the Théodoresco transform is used to show that, under additional assumptions, each Hölder continuous function f defined on the boundary Γ of a fractal domain Ω ⊂ ℝ2n can be expressed as f = Ψ+ − Ψ−, where Ψ± are Hölder continuous functions on Γ and Hermitian monogenically extendable to Ω and to ℝ2n ∖ (Ω ∪ Γ) respectively.

The connection between Clifford analysis and the Weyl functional calculus for a d-tuple of bounded selfadjoint operators is used to prove a geometric condition due to J. Bazer and D. H. Y. Yen for a point to be in the support of the Weyl functional calculus for a pair of hermitian matrices. Examples are exhibited in which the support has gaps.

Orthogonal Clifford analysis is a higher dimensional function theory offering both a generalization of complex analysis in the plane and a refinement of classical harmonic analysis. During the last years, Hermitean Clifford analysis has emerged as a new and successful branch of it, offering yet a refinement of the orthogonal case. Recently in [F. Brackx, B. De Knock, H. De Schepper, D. Peña Peña, F. Sommen, submitted for publication], a Hermitean Cauchy integral was constructed in the framework of circulant (2×2) matrix functions. In the present paper, a new Hermitean Hilbert transform is introduced, arising naturally as part of the non-tangential boundary limits of that Hermitean Cauchy integral. The resulting matrix operator is shown to satisfy properly adapted analogues of the characteristic properties of the Hilbert transform in classical analysis and orthogonal Clifford analysis.

Hermitean Clifford analysis is a higher dimensional function theory centered around the simultaneous null solutions, called Hermitean monogenic functions, of two Hermitean conjugate complex Dirac operators. As an essential step towards the construction of an orthogonal basis of Hermitean monogenic polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is established for such polynomials. The minimal number of initial polynomials needed to obtain a unique Hermitean monogenic extension is determined, along with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya extension principle then allows for a dimensional analysis of the spaces of spherical Hermitean monogenics, i.e. homogeneous Hermitean monogenic polynomials. A version of this extension theorem for specific real-analytic functions is also obtained.

We consider a left-linear analogue to the classical Riemann problem: Dau =0 inRnn u+ = H(x)u +h(x )o n ju(x)j = O(jxj n 2 1 )a sjxj!1: For this purpose, we state a Borel-Pompeiu formula for the disturbed Dirac operatorDa = D +a with a paravectora and some functiontheoretical results. We reformulate the Riemann problem as an integral equation: Pau +HQau = h on ; where Pa = 1 2(I +Sa )a ndQa = I Pa: We demonstrate that the essential part of the singular integral operator Sa which is constructed by the aid of a fundamental solution of D +a is just the singular integral operator S asso- ciated toD: In case Sa is simplyS and = Rn 1, then under the assumptions 1.H= P He and allH are real-valued, measurable and essentially bounded; 2. (1 +H(x))(1 +H(x)) and H(x) H(x) are real numbers for all x2Rn 1; 3. the scalar part H0 of H fulls H0(x) >"> 0 for all x2Rn 1; the Riemann problem is uniquely solvable in L2;C(Rn 1) and the successive approximation