Fractional integration

Summary

Fractional integration on $\R^n$ is the operator $I_\alpha$, for $0 < \alpha < n$, that convolves a function with the kernel $\abs{x}^{\alpha - n}$, suitably normalized. It is also called the Riesz potential. On the Fourier transform side it multiplies by a constant times $\abs{\xi}^{-\alpha}$, so it is the inverse of the fractional Laplacian $(-\Delta)^{\alpha/2}$. It is the Euclidean analogue of the Riemann–Liouville integral, and it is the main tool behind the Hardy–Littlewood–Sobolev and Sobolev inequalities.