I conjecture that if the functions $f$, $g$ defined on $\mathbb{R}^n$ satisfying $$|f(x)| ≤ A(1+|x|)^{−M}, \quad |g(x)| ≤ B(1+|x|)^{−N}$$for some $M$, $N > n$, then$$|(f * g)(x)| ≤ ABC(1+|x|)^{−L},$$where $L = \min(N,M)$ and $C =C(N,M) > 0$.
From playing around, I've had the idea 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, as$$\int_{\mathbb{R}^n}(1+|y|)^{M−N}dy$$is not integrable if $M-N \ge -n$.
How do we fix the problem?
