Laplacian decomposition of vector fields on fractal surfaces

In the present paper we consider domains in R3 with fractal boundaries. Our main purpose is to study the boundary values of Laplacian vector fields, paying special attention to the problem of decomposing a Holder continuous vector field on the boundary of a domain as a sum of two Holder continuous vector fields which are Laplacian in the domain and in the complement of its closure, respectively. Our proofs are based on the intimate relationships between the theory of Laplacian vector fields and quaternionic analysis.