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 Holder-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.