If we know that on [x₀, x₀ + Δx) function f is always inside [a, b), and that on [x₁, x₁ + Δx) function g is always inside [c, d), what can we say about that interval's contribution to their convolution (f * g) on [x₀ - x₁ - Δx, x₀ - x₁)? (I’m assuming here that this is the right interval, and I might have gotten it a bit wrong.)
I think that the contribution must necessarily be in [acΔx, bdΔx), if we assume that all of (a, b, c, d) are positive. (The other cases are important to cover in practice but conceptually don’t matter much.) This gives us both an upper and a lower bound. We can extend this to the case of infinite Δx specifically and only in the case where either a = b = 0 or c = d = 0. Applying this approach to all possible intervals allows us to bound the convolution of two functions by working from an interval-wise decomposition of them.
It might be useful also to know the integrals of the functions over the relevant intervals, or at least bounds on them. The contribution to the convolution cannot be more than the product of the integrals (again, assuming all positive).