The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a [stereographic projection](http://mathworld.wolfram.com/StereographicProjection.html) onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere $\mathbb{S}^N$.) Namely, letting 
$$
\Omega=\frac{2}{1+\lvert x \rvert^2},\qquad \left(x\in \mathbb{R}^N\right)
$$
denote the conformal factor of the stereographic projection, one has the formula 
$$\tag{1}
-\Delta\_{\mathbb{R}^N}\phi = \Omega^{\frac{2+N}{2}} \left(-\Delta\_{\mathbb{S}^N} + \frac{N(N-2)}{4}\right)\left(\Omega^{1-\frac{N}{2}}\phi\right).
$$
My question is: does there exist an analogue of (1) for the fractional Laplacian $(-\Delta\_{\mathbb{R}^N})^\alpha$? Actually I am interested in the cases $\alpha=\pm \frac{1}{2}$. 



Fractional Laplacian and stereographic projection
Unless I'm misreading, there exist such an analogue at least for some $\gamma$. See equation 2.5 and theorem 3.2 of [Fractional Laplacian in Conformal Geometry](http://www.pagines.ma1.upc.edu/~mgonzalez/Papers/Chang_Gonzalez.pdf) by Sun-Yung Alice Chang and María del Mar Gonzáles.


To spell it out, the theorem 3.2 says that for $\gamma \in (0,\frac{n}{2})\setminus \mathbb{N}$ and a smooth function $f:\mathbb{R}^n\to \mathbb{R}$ one has
$$
P\_\gamma[g\_\mathbb{H},|dx^2|] f = (-\Delta\_x)^\gamma f,
$$
where $P\_\gamma[g^+,\widehat{g}]$ is the fractional power of the conformal Laplacian with power $\gamma$ associated to the ambient metric $g^+$ and the conformal class $[\widehat{g}]$. The metric $g\_\mathbb{H}$ is the standard hyperbolic metric of the upper space and $|dx^2|$ is the standard Euclidean metric.


Now if we take the rescaled metric $\widehat{g}\_v = v^{\frac{4}{n-2\gamma}}\widehat{g}$,
equation 2.5 claims that
$$
P\_\gamma[g^+,\widehat{g}\_v] \phi = v^{-\frac{n+2\gamma}{n-2\gamma}}P\_\gamma[g^+,\widehat{g}] (v \phi),
$$
for all smooth functions $\phi$. 


The conformal fractional Laplacian $P\_\gamma[g\_\mathbb{H},v^{\frac{4}{n-2\gamma}}|dx^2|]$ plays the role of the spherical conformal Laplacian in the classical formula (1) of OP.


