I'm trying to understand the concept of fractional Laplacian, and I found the page https://www.ma.utexas.edu/mediawiki/index.php/Fractional_Laplacian，and the formula $$(−\Delta)^sf(x)=c_{n,s}\int_{\mathbb{R}^n}\frac{f(x)−f(y)}{|x−y|^{n+2s}}dy$$ is given there.
Now I'm very confused. Even if $f$ is bounded, the integral may be infinity, unless the integral is considered as the Cauchy principle value. I think if $f$ is regular enough, one doesn't need to worry about the singularity.
However, what bothers me is the case $f$ itself is not integrable in $\mathbb{R}^n$. For example, let $n=1, f(x)=x$, then $f$ is smooth, but can we really define the fractional derivative by the formula above?
Can anyone give me a good reference to start with? I'm confused by a lot of concept actually. Thanks in advance!
Let $0<s<1$ be fixed, and let $\alpha>0$ be such that $\alpha>2s$. Suppose $f\in C^{0,\alpha}(\mathbb{R}^{n})$ and
$$|f(x)|\lesssim (1+|x|)^{\delta},\quad x\in\mathbb{R}^{n}$$
for $0<\delta<2s$. Then I claim that the integral above is absolutely convergent. Indeed, for
\begin{align*} &\int_{1\geq |x-y|\geq\epsilon}\dfrac{|f(x)-f(y)|}{|x-y|^{n+2s}}dy+\int_{|x-y|>1}\dfrac{|f(x)-f(y)|}{|x-y|^{n+2s}}dy\\ &\lesssim\int_{1\geq|x-y|\geq\epsilon}\dfrac{|x-y|^{\alpha}}{|x-y|^{n+2s}}dy+\int_{|x-y|\geq 1}\dfrac{|f(x)|}{|x-y|^{n+2s}}dy+\int_{|x-y|\geq 1}\dfrac{|y|^{\delta}}{|x-y|^{n+2s}}dy\\ \end{align*}
The second integral clearly converges. The third integral converges since for $|y|\geq 2|x|$, $|x-y|\geq|y|/2$, and $n+2s-\delta>n$ by hypothesis. For the first integral, $n+2s-\alpha<n$, so the limit exists as $\epsilon\rightarrow 0$.