Fourier transform of a product in two dimension I am actually interested in a more specific case than the title.
Let Fourier transform of $f(\vec{x}^2)$ be $\mathcal{F}[f](t)$:
More specifically,
$\mathcal{F}[f](t) = \int \frac{d^2 \vec{x}}{(2\pi)^2} \exp(-i\vec{x}\cdot\vec{t}) f(\vec{x}^2)$.
(Note that I have intentionally wrote $f(\vec{x}^2)$ to show that f is a function that depends only on the magnitude of $x$).
If I do know the form of $\mathcal{F}[f](t)$, what is then $\mathcal{F}[f \times \ln \frac{\vec{x}^2}{k^2}]$? 
where $\times$ is just a trivial product and $k$ is a constant.
I suppose this is some general two-dimensional convolution theorem, but I am not really exactly sure about the specific form of the generalized convolution theorem or the fourier transform of $\ln \frac{\vec{x}^2}{k^2}$
 A: It is well known that $-\frac1{4\pi}\ln(x^2+y^2)$ is a fundamental solution of the Laplace equation on the plane and that it's inverse Fourier transform "should be" $|\xi|^{-2}$. Note that both functions are not integrable in $\mathbb R^2$. In Vladimirov V.S., Equations of mathematical physics, $\S9.7$, distribution
$\cal P\frac1{|x|^2}$, 
$$
\left(\cal P\frac1{|x|^2},\varphi\right)=
\int_{|x|<1}\frac{\varphi(x)-\varphi(0)}{|x|^2}\,dx+
\int_{|x|>1}\frac{\varphi(x)}{|x|^2}\,dx
$$
is discussed and proven that
$$
F\left(\cal P\frac1{|x|^2}\right)=-2\pi \ln |\xi|-2\pi C_0, 
$$
where $C_0=\gamma -\log (2)$.
The FT is defined there as
$$
\int e^{i \xi x} f(x)\,dx
$$
So one should modify constants accordingly. 
From there it is possible to express $\ln|\xi|$, find $\mathcal F[\ln|\xi|]$ and get 
$$
\mathcal{F}[f \times \ln \frac{\vec{x}^2}{k^2}]=
\mathcal{F}[f \times \ln \vec{x}^2]-\ln k^2\mathcal{F}[f]=
$$
$$
\mathcal{F}[f ]*\mathcal{F}[\ln|x|]-\ln k^2\mathcal{F}[f].
$$
The resulting formula should work at least for smooth functions $f$ with compact support.
