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On the notion of the Bochner-Martinelli integral for domains with rectifiable boundary

Abstract:

In this paper we discuss the notion of the Bochner-Martinelli kernel for domains with rectifiable boundary in ℂ2, by expressing the kernel in terms of the exterior normal due to Federer (see [17,18]). We shall use the above mentioned kernel in order to prove both Sokhotski-Plemelj and Plemelj-Privalov theorems for the corresponding Bochner-Martinelli integral, as well as a criterion of the holomorphic extendibility in terms of the representation with Bochner-Martinelli kernel of a continuous function of two complex variables. Explicit formula for the square of the Bochner-Martinelli integral is bkp_rediscovered for more general surfaces of integration extending the formula established first by Vasilevski and Shapiro in 1989. The proofs of all these facts are based on an intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis. © 2007 Birkhäuser Verlag Basel/Switzerland.