I am quite interested in how to use fractional Laplacians to characterize derivatives of a (let's say smooth) function. Consider $f$ is defined on torus $\mathbb{T}^n$ and is mean-zero. Since $-\Delta$ has a strictly positive, discrete spectrum for mean-zero $L^2$ functions, we may define fractional Laplacians $(-\Delta)^s$ spectrally. From a purely regularity perspective, this is an order-$2s$ operator, and these operators seem to be basically the same as $D^{2s}$ when you work with $L^2$ based Sobolev spaces.
However, I am pretty sure that there are key differences between $(-\Delta)^{1/2}f$ and $\nabla f$ pointwisely, as the former one is nonlocal but the latter one is local. What is the key difference between these two objects? Moreover, while $\nabla f(x) = 0$ at some point $x$ means that $f$ has a critical point, what does $(-\Delta)^{1/2}f(x) = 0$ at $x$ mean?
Thank y'all in advance!