I am reading a paper where there is the following: $$\mathcal{F}(f)(k)=a|k|\mathcal{F}(g)(k)$$ where $\mathcal{F}(f)(k)$ is the Fourier Transform of the function $f(x)$ to the new variable $k$. The paper claims that upon inverting one gets: $$f(x) = -\frac{a}{\pi}\int_{-\infty}^{+\infty}\frac{g(x-y)-g(x)}{y^2}{\rm d}y$$ I understand that (provided all the function are nice as required) in the first equation, i can set $\mathcal{F}(h)(k)\equiv |k|$ such that I get $$\mathcal{F}(f)(k)=a\mathcal{F}(g*h)(k)$$ leading to $$f(x)=a(g*h)(x)$$ with $h=\mathcal{F}^{-1}(|k|)(x)=-\sqrt{\frac2\pi}\frac{1}{x^2}$ which should lead to $$f(x) = -a\sqrt{\frac2\pi}\int_{-\infty}^{+\infty}\frac{g(x-y)}{y^2}{\rm d}y$$
so my question is, what is the other term in the integrand of the second equation doing and how do i get it? (I assume the difference of $1/\sqrt{2\pi}$ might be due to different fourier transform definition used by the authors and do not really bother me as much as $\int_{-\infty}^{+\infty}\frac{g(x)}{y^2}{\rm d}y$)