The paper considers the fractional Hilbert transform and the fractional Dirac operator based on the properties of Riesz fractional integro-differentiation. After giving the basics of Riesz fractional integro-differentiation in Section 3, the fractional Hilbert operator \(\mathcal{H}^{\alpha}\) is defined in Section 4 as a linear combination of a multiplication with constants and an application of the Hilbert operator \(\mathcal{H}\) and it is proved in Theorem 4.3. that \(\mathcal{H}^{\alpha}\) is an invertible operator on \(L^p(\mathbb R^n)\), \(1<p<\infty\).
The fractional Dirac operator is then defined in Section 5 as \(D^{\alpha}=\mathcal{H}^{\alpha}(-\Delta)^{\alpha/2}\).
For the entire collection see [Zbl 1350.47001].