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 $\ln \frac{\vec{x}^2}{k^2}$
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.