Connection between filter equation in frequency domain and difference equation in time domain?
Hint:

The transfer function $H(\omega)$ is equal to the ratio of output and input spectra:

$$H(\omega)=\frac{Y(\omega)}{X(\omega)}$$

So your first equation can be rewritten as

$$Y(\omega)=X(\omega) - 1.176X(\omega)e^{-j\omega} + X(\omega)e^{-2j\omega}$$

What do you get if you transform this equation to the time-domain?