A metric dependent Hilbert transform in Clifford analysis (Q2469061)

The authors start with an introduction into Clifford analysis in a metric dependent setting: With a metric tensor \(G=(g_{ik})_{i,k=0,\ldots,m}\) they assume a covariant basis \((e_i)\) and a contravariant one \((e^k)\) with suitable defining rules for the multiplication of the basis elements. A vector resp. the anisotropic Dirac operator are given by \[ x= \sum_{k=0}^m e_kx^k\;\; resp.\;\; \partial_g = \sum_{k=0}^m e^k\partial_{x^k} . \] They then are able to define a corresponding Cauchy integral and to prove Plemelj-Sokhotski formulas. This gives rise to an anisotropic Hilbert transform whose properties are studied. The associated Cauchy transform is no longer uniquely determined.