convolution of two decaying polynomial 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?


If $|y|< \frac{|x|}{2}$ then $|x-y| \geq |x|-|y|>\frac{|x|}{2}$

$$|(f \ast g)(x)| \leq \int_{\Bbb{R}^n}|f(y)||g(x-y)|dy$$ $$=\int_{|y| \geq \frac{|x|}{2}}|f(y)||g(x-y)|dy + \int_{|y| < \frac{|x|}{2}}|f(y)||g(x-y)|dy$$ $$\leq AB \int_{|y| \geq \frac{|x|}{2}}\frac{1}{(1+|y|)^M}\frac{1}{(1+|x-y|)^N}dy +AB\int_{|y| < \frac{|x|}{2}}\frac{1}{(1+|y|)^M}\frac{1}{(1+|x-y|)^N}dy$$
$$ \leq AB \frac{2^M}{(2+|x|)^M}\int_{|y| \geq \frac{|x|}{2}}\frac{1}{(1+|y-x|)^N}dy+AB \frac{2^N}{(2+|x|)^N}\int_{|y| < \frac{|x|}{2}}\frac{1}{(1+|y|)^M}dy$$
$$\leq AB \frac{2^M}{(1+|x|)^M}\int_{\Bbb{R}^d}\frac{1}{(1+|y-x|)^N}dy+AB \frac{2^N}{(1+|x|)^N}\int_{\Bbb{R}^d}\frac{1}{(1+|y|)^M}dy$$
$$\leq AB \frac{2^M}{(1+|x|)^L}\int_{\Bbb{R}^d}\frac{1}{(1+|y|)^N}dy+AB \frac{2^N}{(1+|x|)^L}\int_{\Bbb{R}^d}\frac{1}{(1+|y|)^M}dy$$
