Let $f, g: \mathbb R \to \mathbb R$ be nice functions (so everything below make sense ) and let $g_t(x)=g(x/t)$ for $t>0.$
Can we say that $$\sup_{t>0} (g_t\ast f)(x)= \left(\sup_{t>0} g_t \right) \ast f (x)$$? (Here, $\ast$ is the standard convolution) If not, what we can expect?