Given bounded positive functions $f, g, h \in L^1$, with $g, h$ finitely supported, I would like to compare the following two expressions: $$\begin{aligned} a_1 &= \int_{0}^{\infty} \! dx \, f(x) \int_{-\infty}^{\infty} \! dy \, g(y) \int_{-\infty}^{\infty} \! dz \, h(z) \, \frac{\sin[x \, (y-z)]}{\pi \, (y-z)} \\ a_0 &= \int_{0}^{\infty} \! dx \, f(x) \int_{-\infty}^{\infty} \! dy \, g(y) \, h(y) \end{aligned}$$ In other words, $a_0$ is $a_1$ with the sinc kernel replaced by a delta distribution, so an alternative formulation is to look at the behaviour of $$ a_\epsilon = \int_{0}^{\infty} \! dx \, f(x) \int_{-\infty}^{\infty} \! dy \, g(y) \int_{-\infty}^{\infty} \! dz \, h(z) \, \frac{\sin[\tfrac{x}{\epsilon} \, (y-z)]}{\pi \, (y-z)} \\ $$ as $\epsilon$ goes to zero.

I am interested in conditions for when something can be said about the relative values of $a_0$ and $a_1$. Ideally, there would be some bounds for $a_0$ in terms of $a_1$, but even just conditions for when there is a simple inequality would be helpful.