Where did the boundary conditions go in the frequency-space solution to the Green's function for a damped harmonic oscillator?

In differential equations, Green’s functions are only defined given boundary conditions. In fact, you need two of them for a second order differential equation.

In a lot of physics lecture notes, a solution to the Green’s function is solved by going to the Fourier domain. For example, for the damped harmonic oscillator,

$$left( frac{d^2}{dt^2} + gamma frac{d}{dt} + omega_0^2 right) G(t,t’) = delta(t-t’),$$

which we can transform into the Fourier domain by asserting $G(t,t’) = int frac{d omega}{2 pi} e^{-i omega t} G(omega,t’)$, and using the property $delta(t-t’) = int frac{d omega}{2 pi} e^{-i omega (t-t’)}$, to become

$$ (-omega^2 – i gamma omega + omega_0^2) G(omega, t’) = e^{+i omega t’}$$

And therefore

$$ G(omega, t’) = frac{e^{+i omega t’}}{-omega^2 – i gamma omega + omega_0^2},$$

and so

$$ G(t,t’) = int_{-infty}^{infty} frac{d omega}{2 pi} e^{-i omega(t-t’)} frac{1}{-omega^2 – i gamma omega + omega_0^2},$$

which we can solve using a contour integral for $t > t’$ and $t < t’$. The poles are both in the LHS, which results in causality, i.e. a factor of $Theta (t)$, the Heaviside function.

However, boundary conditions were not once invoked in this derivation, yet alone two. What gives? Does one of the steps sneakily contain boundary conditions?