Cauchy transform on nonrectifiable surfaces in Clifford analysis

The problem of reconstructing a monogenic Clifford algebra valued function on the boundary Γ of a general open set Ω in R^{n + 1} from a prescribed jump data u over the boundary is deeply connected with the study of the Clifford-Cauchy transform(C_Γ u) (x) : = under(∫, Γ) frac(over(y - x, -), σ_n | y - x |^{n + 1}) n (y) u (y) d y, x ∉ Γ . Necessary and sufficient condition on non-rectifiable Γ is established guaranteeing the existence of continuous boundary values of this transform for all functions satisfying a Hölder type condition.