Hölder norm estimate for the Hilbert transform in Clifford analysis (Q711427)

For a Jordan domain \(\Omega \in \mathbb{R}^n\) with \(d\)-summable boundary \(\Gamma\) the definitions of Cauchy and Hilbert transforms are given using Clifford analysis and the Whitney extension of a Hölder continuous function on a compact set \(E\). The authors first prove that the Cauchy transform can be extended continuously to \(\overline{\Omega}\) and that the Hilbert transform is Hölder continuous. The norm of the Hilbert transform is then estimated by an explicit expression containing the diameter of \(\Gamma\), \(d\), and the \(D\)-sum of \(\Omega\) and \(\Gamma\), where \(D\) depends on \(n\) and \(d\) in a not so simple manner. The result gives also improvements for the complex plane.