Let $(-\Delta)^s $ be the fractional laplacian. Consider the space dimension to be $n = 1$, so that
$$ (-\Delta)^s u = C_{1,2s}p.v.\int_{\mathbb{R}}\frac{u(x)-u(y)}{|y-x|^{1+2s}}dy $$
Do you know some literature about the case $n < 2s$, namely $n = 1$ and $s\in(\frac{1}{2},1)$? I want to know the properties of this operator for that range of values since most people focus on the case $n > 2s$ due to the Riesz kernel's existence (I guess). Thank you for your attention.
There is literature on the case $n =1$. For example, if $n=1$, $\frac 1 2 <s<1$ then the fundamental solution is still given by the Reisz potential. If $n=1$ and $s= \frac 1 2 $ then the fundamental solution is $- \frac 1 \pi \log \vert x \vert$.
For further reading see Fractional Thoughts by Nicola Garofola, in particular, Theorem 8.4 and Remark 8.7. In Remark 8.7 he also references another paper which may be of use to you, but I haven't read this paper myself, so I can't confirm that.
