Fractional Laplacian on Euclidean space

For $\alpha>0$ , the fractional Laplacian on $\mathbb R^n$ is defined first for Schwartz functions by

\widehat{(-\Delta)^\alpha f}(\xi)=(2\pi|\xi|)^{2\alpha}\widehat f(\xi),

using the Fourier convention $\widehat f(\xi)=\int e^{-2\pi i x\cdot\xi}f(x)\,dx$ . The power times $\widehat f$ defines a tempered distribution, so distributional Fourier inversion defines the output. At $\alpha=1$ this is the negative Laplacian. The notation $|\nabla|^\beta$ means $(-\Delta)^{\beta/2}$ .

The constant $\pi$ appears because frequencies are measured in cycles per unit length.

Positive quadratic form

Plancherel gives $\langle f,(-\Delta)^\alpha f\rangle=\int(2\pi|\xi|)^{2\alpha}|\widehat f|^2\geq0$ . For noninteger $\alpha$ the symbol need not be smooth at zero, so this formula is not a license to multiply every tempered distribution by the symbol. On $L^2$ , the natural operator domain consists of functions for which the displayed product belongs to $L^2$ .