In different places I've seen Schwarz functions defined by $$\sup_{|\alpha|\le N}\sup_{x\in\mathbb{R}^n}\left(1+|x|\right)^N|D^\alpha \psi(x)| < \infty$$ and by $$\sup_{|\alpha|\le N}\sup_{x\in\mathbb{R}^n}\left(1+|x|^2\right)^N|D^\alpha \psi(x)| < \infty.$$
I wish to show these are equivalent conditions. That the latter implies the first is easy: both expressions are bounded in the unit ball, so we only need to check how they behave outside of it, and $$(1+|x|)\le (1+|x|^2)$$ outside the unit ball.
How could one show the other direction?
It follows by taking $2N$ in the first criterion and the fact that $$ (1+|x|)^{2N} = (1 + 2\,|x|+|x|^2)^N \geq (1+|x|^2)^N. $$