A note on the Bochner–Martinelli integral

Abstract: Let Ω be a simply connected bounded domain in 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 integralhas continuous limit values on Γ if the truncated integrals. 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 obtained for the context of piece-wise Liapunov surface of integration. [Copyright &y& Elsevier]