For a "good" function $u$, I consider its (Gagliardo) fractional Laplacian ($0<s<1$) $$ (-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy, $$ at least as a principal value and up to a constant. I wonder if there is a representation formula in the particular case $u=u(|x|)$, i.e. when $u$ is a radially symmetric function. In particular, is there any relationship between $(-\Delta)^s u$ and $(-\Delta)^s v$, where $v(|x|)=u(|x|^\beta)$, $\beta >0$?

All these questions have trivial answers when $s=1$, and I suspect that at least partial answers can be given also for $0<s<1$, but I cannot find any good reference.