Suppose I have a smooth function $\psi$ from $\mathbb{R}^n$ to $\mathbb{C}$, for which I know that $$ \sup_{x\in\mathbb{R}^n}\left||x|^k\Delta^{p}\psi(x)\right|<\infty $$ for all $k,p\in\mathbb{N}_0$. Is this already sufficient for $\psi$ to be a Schwartz function?
I have tried fooling around with the Fourier transform, the problem seems to be however, that I am not able to control terms of the form $$ |x|D^p(|x|^{2k}\hat{\psi}) $$ I am quite sure that the solution is pretty elementary, but I cannot figure it...