The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection 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 $\alpha=\pm \frac{1}{2}$.