$f\in W^{1,2}(\mathbb R)$ and $\|xf(x)\|_2<\infty$ implies $\lim_{x\to\infty}x|f(x)|^2=0$

Question

I'm trying to apply an integration by parts to solve the Exercise 8.18 at Folland's Real Analysis. But, for that, I need to have "$f\in W^{1,2}(\mathbb R)$ and $\|xf(x)\|_{L^2}<\infty$ implies $\lim_{x\to\infty}x|f(x)|^2=0$".

I couldn't solve it. I tried to use Morrey's inequality but I couldn't figure it out how to handle that limit. I'd be glad for any help.

Answer 1

I've just found Folland's article related to this topic. It contains the explanation. Here is the link if anyone wants to read: http://www.math.stonybrook.edu/~bishop/classes/math533.S21/Notes/Folland_uncertainty.pdf