I am trying to solve the equation $(-\Delta)^s \varphi(x) = f(x)$, $x\in \mathbb{R}^3$ for $\varphi$ and $f$ that are spherically symmetric. Therefore, I am wondering if there is a nice simple formula for the operator $(-\Delta)^s$ where $\Delta$ is the Laplacian and $s> 1$, $s\in\mathbb{R}$, in spherical coordinates in 3 dimensions (similar to the standard Laplacian). This would simplify my problem significantly.
Here $(-\Delta)^s$ is defined via the Fourier transform on $\mathbb{R}^n$, i.e., for a function $u:\mathbb{R}^n\to \mathbb{R}$, I mean $\widehat{(-\Delta)^s u}(k) = |k|^{2s}\hat{u}(k)$.
The fractional Laplacian of a radial function is given in Lemma 7.1 of Fractional Thoughts by Nicola Garofalo. The formula is by no means as simple as in the case of the Laplacian, but I don’t believe it is possible to come up with a simpler expression than this one.