Show that if the functions $f,g$ defined on $\mathbb{R}^n$ satisfying $|f(x)| ≤ A(1+|x|)^{−M} $and $|g(x)| ≤ B(1+|x|)^{−N}$ for some $M,N > n$,
then $|(f ∗g)(x)| ≤ ABC(1+|x|)^{−L} $, where $L = min(N,M)$ and $C =C(N,M) > 0$.
There is a hint that the inequality $(1+|x-y|)^{-k}\le (1+|y|)^k(1+|x|)^{-k}$ might be used to estimate the decay of this convolution.
I currently have $|(f ∗g)(x)| ≤ AB\int_{\mathbb{R}^n}(1+|x-y|)^{−M}(1+|y|)^{−N}dy \le AB(1+|x|)^{-M}\int_{\mathbb{R}^n}(1+|y|)^{M−N}dy$
But there is a problem $\int_{\mathbb{R}^n}(1+|y|)^{M−N}dy$ is not integrable if $M-N \ge -n$
How to fix the problem?