This is not a homework, and I have no idea whether it could be one. It is only a request for help, as I do not have any experience using Fourier transform. The origin of the problem is from physics. If this kind of more applicative question is inappropriate, please let me know.
I have the following convolution product: $g=-h*f$ where $h(x)=Ax\ /\ (x^2+B)^{3/2}$ with $A, B, l\gt 0$ and $x\notin[0,l] \Rightarrow f(x)=0$
I wish to inverse the definition, i.e., to get the function $f$ explicited, or at least to know whether there are specific constraints for my question to have meaning.
I guess one way to do it is to use the inverse $1/\hat{h}$ of the Fourier transform $\hat{h}$ of $h$. But I do not trust I can do that without error.
In practice, I suppose this can be done numerically with FFT, but I would think one first has to check that it makes sense formally.
I am also wondering how well $f$ can be approximated if the function $g$ is known only on $[0,l]$.
The deconvolution problem you want to solve is typically ill-posed. In principle you are correct that you only have to divide by the Fourier transform of the kernel, but since this will go to zero for high frequencies, inversion will amplify noise there.
However, this may not be a real problem when you have some a-priori knowledge of your function. e.g. when you know that the function is smooth (then it has no frequency components at high frequencies). There are so called regularization schemes which handle these.
One possibility of implementing such regularization or a-priori information, is if you assume the original function $f$ is representable by a e.g. spline, and then solve for the spline coefficients so that the convolution of the spline best matches your result $g$. Then you also only get a linear equation system, which is well solvable with a computer.
There is also Tikhonov regularization.
The FFT approach has another difficulty, since the FFT implements the discrete Fourier transform which is not the continuous Fourier transform, which you need. The discrete Fourier transform has this periodicity assumption in both original domain and frequency domain, which is probably not corresponding to your physical situation. If it does, the FFT is right choice, though.