Suppose that we have a function $f$ that is of exponential type, and that the function $|f(x-i)|^2 \in A_2$ is such that it satisfies $$\left(\frac{1}{|B|} \int_B |f(x-i)|^2\right)\left(\frac{1}{|B|} \int_B \frac{1}{|f(x-i)|^2}\right) \leq C < \infty,$$ for all balls $B$. Can I conclude that there is some lower bound for $|f(x-i)|$? Apparently there is a way to bound it in the sense of $$ |f(x-i)| \geq \frac{C}{1 + |x|^3},$$ for all $x \in \mathbb{R}$.
How is this possible? I assume we would have to proceed by contradiction somehow and essentially arrive to the impossible scenario where the inequality above is false for some appropriately chosen $B$. But I am not sure how. Any hint is welcomed!
Edit: This claim is from a paper I read, I am more interested in the claim itself and not the result in the paper you can find it on page 8 here.