Fourier transform of function division in time domain
For division there is no equivalent to the duality between multiplication and  convolution. Note that from the existence of the Fourier transforms of $f_1(t)$ and $f_2(t)$, you cannot conclude anything about the existence of the Fourier transform of $f_3(t)=f_1(t)/f_2(t)$. So $f_3(t)$ might not even have a Fourier transform, and if it exists, it cannot be expressed in any simple way in terms of $F_1(\omega)$ and $F_2(\omega)$.