Say we know that the function $f \in L_2$, that is $$\int_0^\infty f^2(t) dt < \infty$$
I am interested in the integral $$I(t) = \int_0^t e^{-\eta (t - \tau)}f(\tau)d\tau$$
in particular, one can view $I(t)$ as the output of the low-pass filter $$\dot{I} + \eta I = f(t)$$
This observation suggests that $I(t) \in L_2$ by Parseval's inequality. As just stated, $I(t)$ is a low-pass filter of $f$, and so should only remove energy stored in the high-frequency modes. However, I have been unable to show that $I(t) \in L_2$ directly by computation of the integral $$\Vert I\Vert_2 = \int_0^\infty I(\tau)^2d\tau = \int_0^\infty \left(\int_0^t e^{-\eta (t-\tau)}f(\tau)d\tau\right)^2$$
I was wondering if there is a simple way to do so, as I have not been able to find any resource online. It seems like something that should be doable using Cauchy-Schwarz, but I am stuck.