A note on the Bochner-Martinelli integral

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 integral14 ^π2∫ _Γ ⁿ¹(ζ) (ζ̄ ₁-q̄ ₁)+ ⁿ²(ζ) (ζ̄ ₂-q̄ ₂)|ζ-q| ⁴f(ζ) d ^H3(ζ),q∉∂Ω,has continuous limit values on Γ if the truncated integrals. 12 ^π2∫ _{Γ{ζ:|ζ-z|≤}} ⁿ¹(ζ) (ζ̄ ₁-z̄ ₁)+ ⁿ²(ζ) (ζ̄ ₂-z̄ ₂)|ζ-z| ⁴(f(ζ)- f(z))d ^H3(ζ),converge uniformly with respect to z on Γ as → 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.