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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]

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]

The aim of this paper is to offer a new definition of Cauchy integral associated with Maxwell equations on 3-dimensional domains with fractal boundaries. This new Cauchy integral leads to several types of integral representation formulas, including the Cauchy representation formula for time-harmonic electromagnetic fields. Our approach is based on the framework of exploiting the close and well-defined connection between the quaternionic analysis and Maxwell equations. [ABSTRACT FROM AUTHOR]

In this note, we consider poly-analytic Cauchy integral operators introduced in a rather natural way on the higher order Lipschitz classes. We prove some boundary properties of the function represented by such an integral operator in the general context of rectifiable Jordan curves. [ABSTRACT FROM AUTHOR]