What does a power of $|\nabla|$ mean in the context of PDE? I've seen the notation $|\nabla|^\alpha f$ used in a PDE setting, where $f$ is some function on $\mathbb R^n$. Could someone tell me what that means?
For example in this discussion on Math Overflow the first answer uses this notation
https://mathoverflow.net/questions/48292/applications-of-hardys-inequality
 A: The only reasonable choices are: the power of $|\nabla f|$, and fractional gradient of $f$. Looking at the source:
$$\left\| \frac{f}{|x|^\alpha} \right\| _ {L^p({\bf R}^n)} \leq C_{p,\alpha,n} \| |\nabla|^\alpha f \|_{L^p({\bf R}^n)}$$
we see that both sides are homogeneous of degree $1$ with respect to $f$. So it's the fractional gradient, then. There are multiple ways to define these, but from the Harmonic analysis perspective one usually wants a Fourier multiplier. Multiplication by $|\xi|^\alpha$ on the Fourier side is a common choice: it has the effect similar to $\alpha$-order differentiation, although it's more precisely $(-\Delta)^{\alpha/2}$ rather than $|\nabla |^\alpha$. 
In particular, $\alpha=1$ does not recover $|\nabla f|$, but is loosely equivalent to it for the purposes of the smoothness/integrability considerations. Specifically, one obtains $\nabla f$ from $(-\Delta)^{1/2}f$ by means of the Fourier multiplier $\xi/|\xi|$, which corresponds to the Riesz transform (in one dimension, it's the Hilbert transform). Riesz transform maps many function spaces,  such as $L^p$ for $1<p<\infty$, to themselves.  So, $\|\nabla f\|_p \approx \|(-\Delta)^{1/2}f\|_p$. 
