Difficulty with Kallen-Lehmann spectral functions

The following is for $D=4$. The correlators at a fixed point are power laws of the form $x^{-2Delta}$, where $Delta$ is the scaling dimension. Suppose I wish to find the nature of the spectrum at the fixed point, for which I calculate the spectral function $rho(p^2)$ which is defined so that $$langlephi(x)phi(0)rangle=int frac{d^4p}{2pi}^4e^{-ipx}rho(p^2)$$

Now, for $Delta=1$, I expect this to be the same as that of a fundamental scalar field, with $rho(p^2)=delta(p^2)$.

$Delta=2$ should correspond to a composite operator of 2 massless fields, and thus I expect $rho(p^2)=int d^4kdelta[(k-p)^2]delta(k^2)$ and so on.

However, I am unable to derive these relations formally. Any help would be appreciated.

As an example of the kind of attempts I made-none of which I claim to be rigorous in any way-note that the problem reduces to finding the fourier transform of $frac{1}{x^2}$ and $frac{1}{x^4}$. I tried introducing a regulator to control $xto 0$, but got nowhere. Another approach was to call $int d^4x e^{ipx}frac{1}{x^4}=f(p)$, and find the differential equation for $f(p)$ by differentiating both sides with respect to $p$ until the LHS reduces to something like $int d^4p e^{ipx}=delta(x)$. It is likely that this is infact the right way to proceed, but I’m a little lost and frustrated by what should have been a simple calculation.