Fourier transform and maximum

I doubt it. The Fourier transform of max(f, g) can be computed efficiently if and only if the Fourier transform of |f| can be computed efficiently. (Because max(f,g) = (f+g+|f-g|)/2.)

But there seems to be no relationship between F{f} and F{|f|}...

Assuming you mean the max at each point, and since max is a non-linear operation, there is not going to be any way to do this. You would need to do the max operation in the time domain and then perform the Fourier transform.