The Bochner–Martinelli transform with a continuous density: Davydov's theorem

Abstract

In this paper, we extend to the theory of functions of several complex variables, a theorem due to Davydov from classical complex analysis. We prove the following: if Ω⊂ℂ ⁿ is a bounded domain with boundary ∂Ω of finite (2n−1)-dimensional Hausdorff measure ℋ²ⁿ⁻¹ and f is a continuous complex-valued function on ∂Ω such that

converges uniformly on ∂Ω as r→0, then the Bochner–Martinelli transform on Ω of f admits a continuous extension to ∂Ω and the Sokhotski–Plemelj formulae hold. For n=2, we briefly sketch how quaternionic analysis techniques may be used to give an alternative proof of the above result.

Keywords:

• Bochner–Martinelli transform
• Sokhotski–Plemelj formulae
• non-smooth boundaries

2000 Mathematics Subject Classifications :

• Primary: 32A26
• Secondary: 32A10

Acknowledgements

This paper was written while the second author was visiting the Department of Mathematical Analysis of Ghent University. He was supported by the Special Research Fund No. 01T13804, obtained for collaboration between the Clifford Research Group in Ghent and the Cuban Research Group in Clifford analysis, on the subject Boundary values theory in Clifford Analysis. Juan Bory Reyes wishes to thank all members of this Department for their kind hospitality. Dixan Peña Peña was supported by a Doctoral Grant of the Special Research Fund of Ghent University. He would like to express his sincere gratitude. The authors acknowledge the valuable suggestions and comments of both referees, which turned out to be helpful in improving this paper.