A fractional Dirac operator. (English) Zbl 1394.47047
Alpay, Daniel (ed.) et al., Noncommutative analysis, operator theory and applications. Selected papers based on the presentations at the conference, Milano, Italy, June 23–27, 2014. Basel: Birkhäuser/Springer (ISBN 978-3-319-29114-7/hbk; 978-3-319-29116-1/ebook). Operator Theory: Advances and Applications 252. Linear Operators and Linear Systems, 27-41 (2016).
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].
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].
Reviewer: Damir Bakić (Zagreb)
MSC:
47F05 | General theory of partial differential operators |
26A33 | Fractional derivatives and integrals |
Keywords:
fractional Dirac operator; fractional Hilbert operator; fractional Laplacian; Riesz potentials; Riesz integro-differentiation; Clifford analysisReferences:
[1] | pp. |
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