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Are the decay rates at infinity of functions inherited when they are convolved?

Suppose for instance that $f$ and $g$ belong respectively to $L^p(\mathbb{R}^n)$ and $L^q(\mathbb{R}^n)$, with $p,q$ such that $\frac{1}{p}+\frac{1}{q} \geq 1$. Suppose further that $|f(x)| = O(|x|^{-\alpha})$ and that $|g(x)| = O(|x|^{-\beta})$ as $|x| \to \infty$.

Is it true that $|f*g(x)| = O(|x|^{-\min \{\alpha,\beta \} }), |x| \to \infty$ ? Can it at least be established for $p=q=1$ ?

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