I need to calculate the Laplace transform $\mathcal{L_t}$ of the function
$$ h(t)=\int_0^{\tau-t}f(\tau-t-u)g(u)du, $$
in terms of $\mathcal{L_t}[(f\ast g)(t)]$. Here, I do not know $g$ and $\tau>t>0$. I am trying to use the property
$$\mathcal{L_t}[(f\ast g)(\tau-t)] = \mathcal{L_t}[f(\tau-t)]\mathcal{L_t}[g(\tau-t)],$$
but doing an example it seems to not be correct. I am also using the properties of Laplace transform of translation and multiplication of sign, but without success. Does someone know an interesting property for this case?