Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other functions $f_1,g_1$ such that for large enough $|x|$, $|f(x)|\geq |f_1(x)|$, and $|g(x)| \geq |g_1(x)|$, what can I then say about the convolution
$$ f*g(x) = \int f(y)g(x-y)dy ?$$
I suspect that I am about to bound from below $|f*g|$ by something like $|f_1(x)g_1(x)|$ or $|f_1(x)| + |g_1(x)|$ but I cannot seem prove it.