The Laplace transform - does it have an associated differential operator, if the kernel is to be viewed as a Green's function?

Question

I've begun learning about Green's functions, and if I understand correctly, the Green's function for a linear differential operator $L$ with appropriate boundary conditions is the kernel for the integral transform which transforms the system input into the system output, so if $$Ly=f(x),$$ then $$y(x)=\int_a^b G(x,t)f(t)dt$$

I immediately thought of the Laplace transform:

$$F(s)=\int_0^\infty e^{-st}f(t)dt$$

If the exponential were to be treated as a Green's function, what differential operator would it correspond to?