A note on the Bochner–Martinelli integral

摘要

Let Ω be a simply connected bounded domain in C2 with boundary an Ahlfors David regular surface Γ and f be a continuous function on Γ. Ahlfors David regular surfaces include a broad range of those from smooth to piece-wise Liapunov and Lipschitz surfaces.Using intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis we prove that the Bochner–Martinelli integral14π2∫Γn1(ζ)(ζ¯1-q¯1)+n2(ζ)(ζ¯2-q¯2)|ζ-q|4f(ζ)dH3(ζ),q∉∂Ω,has continuous limit values on Γ if the truncated integrals.12π2∫Γ⧹{ζ:|ζ-z|⩽ϵ}n1(ζ)(ζ¯1-z¯1)+n2(ζ)(ζ¯2-z¯2)|ζ-z|4(f(ζ)-f(z))dH3(ζ),converge uniformly with respect to z on Γ as ϵ → 0.This allows us to discuss, in the last part of the note, a formula for the square of the singular Bochner–Martinelli integral on Ahlfors David regular surfaces. Our formula is in agreement with that of [18] obtained for the context of piece-wise Liapunov surface of integration.