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 Rn+1 from a prescribed jump data u over the boundary is deeply connected with the study of the Clifford–Cauchy transform (CΓ u)(x) := Γ y − x σn|y −x|n+1 n(y)u(y) dy, 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. © 2007 Elsevier Inc. All rights reserved