Let $f$ be a non-negative function on $\mathbb R^d$ satisfying the following:
(1) There exists a non-increasing function $g$ on $(0,\infty)$ such that
(1-i) $ C_1^{-1} g(|x|) \le f(x) \le C_1 g(|x|) $ for some constant $C_1\ge 1$,
(1-ii) $g(|x|) \le C_2 g(|x|+1), \ |x|\ge 1 $ for some constant $C_2 \ge 1$,
(2) $\int |x|^2 f(x) \ dx < \infty $
I want to claim that $f(x) \le c |x|^{-d-2}$ by the integrability condition, using change of variables (polar coordinate), but can't prove it. I wonder if it does not hold for some $f$.