TY - JOUR
T1 - On the notion of the Bochner-Martinelli integral for domains with rectifiable boundary
AU - Abreu-Blaya, Ricardo
AU - Bory-Reyes, Juan
AU - Shapiro, Michael
N1 - Funding Information:
M. Shapiro was partially supported by CONACyT projects as well as by Instituto Politécnico Nacional in the framework of COFAA and SIP programs.
Funding Information:
The research carried out in this paper was completed during R. Abreu stay as Visiting Professor at IMPA, Rio de Janeiro, supported by CAPES, under the Grant No. 1462-15/2006. He would like to express his sincere gratitude.
PY - 2007/5
Y1 - 2007/5
N2 - 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 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.
AB - 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 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.
KW - Bochner-Martinelli integral
KW - Quaternionic analysis
KW - Rectifiability
UR - http://www.scopus.com/inward/record.url?scp=34247581392&partnerID=8YFLogxK
U2 - 10.1007/s11785-006-0006-6
DO - 10.1007/s11785-006-0006-6
M3 - Artículo
SN - 1661-8254
VL - 1
SP - 143
EP - 168
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 2
ER -