A Fractional Dirac Operator

Abstract

Based on the Riesz potential, S. Samko and coworkers studied the fractional integro-differentiation of functions of many variables which is a fractional power of the Laplace operator. We will extend this approach to a fractional Dirac operator based on the relation $D\;=\;\mathcal{H}(-\Delta)^{(-1/2)}$. Because the Hilbert operator $\mathcal{H}$ is involved as well as the fractional Laplacian of order ${-1/2}$, we will use fractional Hilbert operators and fractional Riesz potentials for the construction.

Mathematics Subject Classification (2010). Primary 26A33; Secondary 30G35