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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.

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

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  • $\begingroup$ 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... $\endgroup$
    – Siminore
    Feb 26, 2014 at 15:02
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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.

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

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