Fractional Laplacian

Remarks

Equivalently, for $u \in \mathcal{S}(\R^n)$ and $x \in \R^n$, $$(-\Delta)^s u(x) = C_{n,s} \lim_{\varepsilon \to 0^+} \int_{\abs{x - y} > \varepsilon} \frac{u(x) - u(y)}{\abs{x - y}^{n + 2s}} \, dy, \qquad C_{n,s} = \frac{4^s \, \Gamma(n/2 + s)}{\pi^{n/2} \abs{\Gamma(-s)}}.$$ The integral formula also makes sense for bounded $C^2$ functions. Example: for $u(x) = \cos(k \cdot x)$ with $k \in \R^n$, the integral formula gives $(-\Delta)^s u = \abs{k}^{2s} u$, in agreement with the multiplier $\abs{\xi}^{2s}$. Non-example: the exponent $s = 1$ is excluded, since the constant $C_{n,s}$ tends to $0$ as $s \to 1$ and the singular integral diverges; the multiplier $\abs{\xi}^2$ gives the ordinary operator $-\Delta$ instead. The function $(-\Delta)^s u$ is smooth but in general not in $\mathcal{S}(\R^n)$, since $\abs{\xi}^{2s}$ is not smooth at $\xi = 0$; it decays like $\abs{x}^{-n-2s}$.