Let $(f*g)(t):=\int_0^t f(s) g(t-s)ds.$
Then the Laplace transform $L$ satisfies $L(f*g)(t)=L(f)(t)L(g)(t).$
This is known as the convolution theorem.
I would like to know whether something similar holds in this situation:
$$F(t):=\int_0^t \int_0^s f(t-s,s-k)u(s)u(k) dk ds.$$
This is still a one-dimensional object, but it looks somehow like a double convolution. Thus, can we say anything about the Laplace transform of this object?
If you have any questions, please let me know.