A note on the Bochner-Martinelli integral

Abstract:

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∫ Γn1(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. © 2011 Elsevier Inc. All rights reserved.

Año de publicación:

2012

Keywords:

• Ahlfors David regular surfaces
• Quaternionic analysis
• Bochner-Martinelli integral

Fuente:

scopus