Abstract
We consider Holder 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 Teodorescu 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.
Highlights
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
Boundary Value Problems e1, . . . , em is an orthonormal basis for the quadratic space R0,m underlying the construction of the real Clifford algebra R0,m, where the considered functions take their values
Since the Dirac operator is invariant with respect to the action of the orthogonal group O m; R, doubly covered by the Pin m group of the Clifford algebra R0,m, the resulting function theory is said to be rotation invariant
Summary
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, referred to as Euclidean Clifford analysis, focuses on the null solutions, called monogenic functions, of the vector-valued Dirac operator ∂X m j1 ej. The resulting function theory focuses on the simultaneous null solutions of two complex Hermitean Dirac operators ∂Z and ∂Z† which no longer factorize but still decompose the Laplace operator in the sense that 4 ∂Z∂Z† ∂Z† ∂Z Δ2n. G12 defined on the fractal which conditions such a the components G12± are boundary Γ of a domain Ω in R2n, and function G12 can be decomposed as G12 extendable to H-monogenic functions in we investigate under G12 − G12−, where the interior and the exterior of Ω, 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.
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