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Let Ω⊂ℝm$$ \Omega \subset {\mathrm{\mathbb{R}}}^m $$ be a bounded regular domain, let ∂x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in ℝm$$ {\mathrm{\mathbb{R}}}^m $$, and let ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space ℝ0,m$$ {\mathrm{\mathbb{R}}}^{0,m} $$. For k∈{1,...,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, ℝ0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}^{(k)} $$ denotes the space of k$$ k $$‐vectors in ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form ∂x_Fk∂x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in Ω$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$‐vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to ∂Ω$$ \mathrm{\partial \Omega } $$, ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator ∂x_(.)∂x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of ℝm$$ {\mathrm{\mathbb{R}}}^m $$. [ABSTRACT FROM AUTHOR]

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in ℝm. The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain Ω⊂ℝm can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for (φ , ψ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ (1 + α , Γ) . [ABSTRACT FROM AUTHOR]

Abstract: Solvability conditions for some Dirichlet type boundary value problems in the framework of Hermitian Clifford analysis are established. [Copyright &y& Elsevier]

Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, calledq-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article. [ABSTRACT FROM AUTHOR]

Suppose that Ω is a bounded domain of ℝ with a fractal boundary Γ and let ℝ be the real Clifford algebra constructed over the quadratic space ℝ. Replacing the fractal dimensions of Γ with conditions of approximating character we will characterize the monogenicity of a ℝ-valued function F in the interior and exterior of Ω, in terms of its boundary value f = F|. Moreover, our geometric perspective allows for generalizations of certain two-sided monogenic extension results to a wide class of domains. [ABSTRACT FROM AUTHOR]

A Hilbert transform for Hölder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ℝ, has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Hölder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Hölder exponents, the diameter of Γ and a specific d-sum ( d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary. [ABSTRACT FROM AUTHOR]

Let Ω ⊂ ℝ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum ( d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface. [ABSTRACT FROM AUTHOR]

We consider Hölder continuous circulant (2 x 2) matrix functions Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. defined on the fractal boundary Γ of a domain Ω in ℝ2n. The main goal is to study under which conditions such a function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. can be decomposed as Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., where the components Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. are extendable to H-monogenic functions in the interior and the exterior of Ω, respectively. H-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. H-monogenic functions then are the null solutions of a (2 x 2) 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: This Note gives geometric conditions on a surface of so that the Hilbert transform on that surface in the framework of Clifford analysis defines a bounded operator in the Hölder continuous functions classes. This result provides a generalization of the well-known theorem of Plemelj and Privalov for curves in . To cite this article: R. Abreu Blaya et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for (φ , ψ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ (1 + α , Γ) . [ABSTRACT FROM AUTHOR]