Fractional Laplacian of radially symmetric functions

For a "good" function $$u$$, I consider its (Gagliardo) fractional Laplacian ($$0) $$(-\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, but I cannot find any good reference.

Theorem 1.1 in http://arxiv.org/pdf/1203.3149.pdf gives an explicit formula.

• I see. However I don't understand how to read that formula: it seems that a rescaling in the radial variable leads to nothing in the fractional laplacian. But is this all we can hope for? I know that we can't use any chain rule, but on the other hand it seems to me that powers of the radial variable $r=|x|$ should lead to a transformation formula... Feb 26 '14 at 15:02

You can also see the formulas (1.6) and (1.7) in http://arxiv.org/pdf/1307.5019.pdf The fractional Laplacian of a radial function is the fractional power of a Bessel operator.

Another representation can be found in Lemma 7.1 in https://arxiv.org/pdf/1712.03347.pdf.