So I know that the Fourier Transform of a function $f(t)$, denoted $\mathcal{F}[f(t)]$ can be used to compute derivatives (and integrals) of $f(t)$, including of non-integer order like so:
$$ \frac{d^n f(t)}{dt^n} = \mathcal{F}^{-1}\{(i\omega)^n \mathcal{F}[f(t)]\} $$
where $\omega$ is the frequency variable. This is known from the properties of the fourier transform, however, I was thinking about how to generalize this for higher-dimensions. Higher dimensional functions $f(\vec r)$ have more interesting differential operators like gradients $\nabla f(\vec r)$ and Laplacians $\nabla^2 f(\vec r)$ and I was wondering how to use the principle stated in the equation above to compute these quantities.
See this blog post of Terry Tao, Remark 2.