A metric dependent Hilbert transform in Clifford analysis

Abstract

In earlier research generalized multidimensional Hilbert transforms have been constructed in $\mathbb{R}^m$ in the framework of Clifford analysis, a generalization to higher dimension of the theory of holomorphic functions in the complex plane. These Hilbert transforms, obtained as part of the boundary value of an associated Cauchy transform in $\mathbb{R}^{m+1}$, might be characterized as isotropic, since the metric in the underlying space is the standard Euclidean one. In this paper we adopt the idea of a so--called anisotropic Clifford setting, leading to the introduction of a metric dependent Hilbert transform in $\mathbb{R}^m$, which formally shows similar properties as the isotropic one, but allows to adjust the co-ordinate system to preferential directions. A striking fact is that the associated Cauchy transform in $\mathbb{R}^{m+1}$ is no longer uniquely determined, but may correspond to various $(m+1)$--dimensional metrics.