A holomorphic extension theorem using Clifford analysis
Let \(\Omega\) be a simply connected bounded and open set in \(\mathbb{C}^n\) and suppose that \(f\) is a \(\mathbb{C}\)-valued continuous function on \(\partial \Omega\). It is a well-known fact that the Bochner-Martinelli integral \(M_1f\) of \(f\) is a harmonic function in \(\mathbb{C}^n \setminus \partial \Omega\) and vanishes at infinity. \textit{A. M. Aronov} and \textit{A. M. Kytmanov} [Funct. Anal. Appl. 9, 254--255 (1976); translation from Funkts. Anal. Prilozh. 9, No. 3, 83--84 (1975; Zbl 0319.32008)] provided the following two characterizations. If \(\partial \Omega\) is a smooth surface and \(f\) is a continuously differentiable function in \(\partial \Omega\), then a necessary and sufficient condition for \(f\) to have a holomorphic extension to \(\Omega\) is that \(M_1f(\underline{x}) = 0\) for \(\underline{x} \in \mathbb{C}^n \setminus \overline{\Omega}\). If \(\partial \Omega\) is a piecewise smooth surface and \(F\) is a continuously differentiable function in \(\overline{\Omega}\), then \(F\) is holomorphic in \(\Omega\) if and only if \(F(\underline{x})=M_1 F(\underline{x})\) for \(\underline{x} \in \Omega\). In this paper alternative characterizations are presented using Clifford analysis, the latter offering a function theory which is a higher dimensional analogue of the theory of the holomorphic functions of one complex variable. The main techniques in obtaining these characterizations are the isotonic functions and the isotonic Cauchy type integral. The results of Aronov and Kytmanov are also re-established for the particular case of \(n=2\) using these techniques.
scientific article; zbMATH DE number 5898471
scientific article