I wish to write the following integral in the frequency domain
$$\int_{-\infty}^{0}\int_{-\infty}^{0} d\tau_2 d\tau_1 f(\tau_1,\tau_2)g(\tau_1,\tau_2) + \int_{0}^{t}\int_{0}^{t} d\tau_2 d\tau_1 f(\tau_1,\tau_2)(g(\tau_1,\tau_2)+h(\tau_1,\tau_2))$$
I have additional information that the functions are time-independent hence $$f(\tau_1,\tau_2) = f(\tau_1-\tau_2)$$ and similar for $g$ and $h$.
I want to represent these in the frequence domain. Since the integral limits are broken and not even completely going to $+\infty$, I am a bit confused towards how to proceed.
Let $$f(\tau_{1}, \tau_{2})g(\tau_{1}, \tau_{2})=\int_{\mathbb{R}^{2}}\Big[\hat{f}(\omega_{1}, \omega_{2})*\hat{g}(\omega_{1}, \omega_{2})\Big]e^{i\omega_{1}\tau_{1}}e^{i\omega_{2}\tau_{2}}d\omega_{1}d\omega_{2}$$ So, $$\int_{-\infty}^{0}f(\tau_{1}, \tau_{2})g(\tau_{1}, \tau_{2})d\tau_{1}d\tau_{2}=$$ $$=\int_{-\infty}^{0}\int_{\mathbb{R}^{2}}\Big[\hat{f}(\omega_{1}, \omega_{2})*\hat{g}(\omega_{1}, \omega_{2})\Big]e^{i\omega_{1}\tau_{1}}e^{i\omega_{2}\tau_{2}}d\omega_{1}d\omega_{2}d\tau_{1}d\tau_{2}=$$ $$=\int_{\mathbb{R}^{2}}\Big[\hat{f}(\omega_{1}, \omega_{2})*\hat{g}(\omega_{1}, \omega_{2})\Big]\int_{-\infty}^{0}e^{i\omega_{1}\tau_{1}}e^{i\omega_{2}\tau_{2}}d\tau_{1}d\tau_{2}d\omega_{1}d\omega_{2}=$$ $$=\int_{\mathbb{R}^{2}}\frac{\Big[\hat{f}(\omega_{1}, \omega_{2})*\hat{g}(\omega_{1}, \omega_{2})\Big]}{\omega_{1}\omega_{2}}d\omega_{1}d\omega_{2}$$ Can do the same with the other integral