# Decay of the fundamental solution of fractional laplacian equations

It is well known that the fundamental solution $\Gamma_1$ in $\mathbb{R}^n$ of the Schrödinger operator $-\Delta + 1$ decays exponentially fast, viz. $|\Gamma_1(x)| \leq C_1 \mathrm{e}^{-C_2|x|}$ as $|x| \to +\infty$. I have read some comments in several papers about a different behavior of the fundamental solution $\Gamma_s$, $0<2<1$, of the fractional Schrödinger operator $(-\Delta)^s +1$; it should decay like $1/|x|^{n+2s}$. Unluckily, I cannot find a good reference for this fact, and I wonder if somebody knows a paper or has some hint for the proof.

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I think it is considered not appropriate to cross-post here and simultaneously, verbatim, on Math Overflow. – paul garrett Aug 10 '12 at 14:05
I could not decide whether my question is more appropriate for this site or for the sister site. It is a research-level question, but also of probable interest for people who study mathematics. I've just deleted my question on Math Overflow. But tell me: do you have any answer or suggestion about the question? – Siminore Aug 10 '12 at 15:12
I would look at the equation in Fourier space. – Fabian Aug 10 '12 at 15:52
@Fabian Maybe. I am a bit suspicious, since the problem for the laplacian operator is usually faced by using radially symmetric solutions, for which explicit estimates are easier. But the fractional laplacian is non-local, and radial symmetry does not reduce everything to an ODE. – Siminore Aug 10 '12 at 16:30