Invert convolution rule for the Laplace transform?

Question

For the Laplace transform, there is a rule to handle convolutions: $$\mathcal{L}\{u*v\}=\mathcal{L}\{u\}\cdot\mathcal{L}\{v\}.$$ In Fourier transform, there is a similar formular and furthermore, there is a formular to invert this convolution and multiplication theorem. Is there something similar for the Laplace transform, like $$\mathcal{L}\{u\cdot v\} = f(U * V)$$ for any or certain classes of functions $U, V$ in general or under certain conditions?

Answer 1

Unfortunately, after doing some research, it seems that there is no general, simply form to solve my problem. Thanks for all contributions.