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

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⊂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 [image omitted] 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. [ABSTRACT FROM AUTHOR]