Criteria for monogenicity of Clifford algebra-valued functions on fractal domains

Abstract

Suppose that Ω is a bounded domain with fractal boundary Γ in ${\mathbb R^{n+1}}$ and let ${\mathbb R_{0,n}}$ be the real Clifford algebra constructed over the quadratic space ${\mathbb R^{n}}$. Furthermore, let U be a ${\mathbb R_{0,n}}$-valued function harmonic in Ω and Hölder-continuous up to Γ. By using a new Clifford Cauchy transform for Jordan domains in ${\mathbb R^{n+1}}$ with fractal boundaries, we give necessary and sufficient conditions for the monogenicity of U in terms of its boundary value u = U|_Γ. As a consequence, the results of Abreu Blaya et al. (Proceedings of the 6th International ISAAC Congress Ankara, 167–174, World Scientific) are extended, which require Γ to be Ahlfors-David regular.