Given that, for some $f\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$, we have the Fourier transform of $f$ given as \begin{equation} F(k):=\int_{-\infty}^\infty e^{-ikx}f(x)\text{d}x~~~(k\in\mathbb{R}),\end{equation} is there any known concise expression for $g$ in terms of $f$, or at least an estimate for $|g|$ in terms of $|f|$, where \begin{equation}g(x):=\frac{1}{2\pi}\int_{-\infty}^\infty e^{ikx}|F(k)|\text{d}k~~~(x\in\mathbb{R})?\end{equation} I can't find anything on this problem in the literature I've looked through. Any information would be greatly appreciated!
[Note: I am assuming, indeed, that $g$ is even well-defined; if it is not, can anybody give me a hint/idea as to why?]