According to wikipedia, the fourier transform of $\frac{1}{|x|^{n-1}}$ is $C\frac{1}{|\xi|^1}$, or more generally the fourier transform of $\frac{1}{|x|^{n-k}}$ is $C\frac{1}{|\xi|^k}$ for $0 < k < n$.
I have no idea why that is true but let's take that for granted for now, now suppose I multiply $\frac{1}{|x|^{-n+1}}$ by a compactly supported smooth function that is supported near the origin, then since the tempered distribution is $L^1$ (say $n>1$), the fourier transform of this will then actually be a smooth function, in other word the singularity of $\frac{1}{|\xi|}$ at the origin should get smoothed out somehow.
This seems to be consistent with something my professors have said, that fourier transform takes growth at infinity to growth at zero, and vice versa, however is there a theorem somewhere that makes this statement formal? Preferably one that address how the singularity at zero gets removed on the frequency side by multiplying a compactly supported function on the spatial side?