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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 n is a bounded domain with boundary of finite (2n-1)-dimensional Hausdorff measure H2n-1 and f is a continuous complex-valued function on such that [image omitted] converges uniformly on as r0, then the Bochner-Martinelli transform on of f admits a continuous extension to 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.