Let $w := e^{-x^2}$, and $u$ be such that $$ \| u \|_{1, w} := \|u\|_{w} + |u|_{1,w} := \int_{\mathbb R} ((u(x))^2 + (u'(x))^2) \, w(x) \, dx \leq \infty. $$ I am trying to understand why $$ \int_{\mathbb R} x^2 (u(x))^2 \, w(x) \, dx \leq C \, \| u \|_{1, w}. $$
In the textbook I'm reading (Spectral method, by Shen, Tang and Wang), the proof of this result (Appendix B8) goes as follows: We start by writing, using integration by parts, $$ \int_{\mathbb R} x u^2 w \, dx = \int_{\mathbb R} uu'w \, dx \leq \|u\|_{\omega} |u|_{1,\omega},$$ to deduce $xu^2w \to 0$ as $|x| \to \infty$. Already at this point, why must the boundary terms be $0,$ and why does integrability imply $xu^2 w \to 0$? I can see that we could perform integration by parts on a bounded interval and then let the bounds go to $\infty$, but how do we conclude that the boundary terms, $[u^2 w]^{b}_a$, go to zero?
The rest of the proof consists in using integration by parts again $$ \int_{\mathbb R} (xu)^2 w dx = \frac{1}{2} \int_{\mathbb R} u^2 w dx + \int_{\mathbb R} x u u' w dx $$ and then Cauchy-Schwarz inequality. Here again, I don't understand exactly how integration by part is applied.
Any clarification would be greatly appreciated.
EDIT: @Gio67 recalled below that integrability and continuity alone do not imply convergence to 0 as $|x| \to \infty$, and therefore the proof in the textbook is not strictly speaking correct. Using his method, one can prove that, for suitably chosen $a_n \to -\infty$ and $b_n \to \infty$, $$ \lim_{n \to \infty} \int_{a_n}^{b_n} x u^2 \, dx = 0, $$ but it's still not clear how one can go from this to the conclusion that $x u^2$ is integrable. It is possible split the domain into the regions $x \geq 0$ and $x < 0$ and to use monotone convergence, but this requires to know that $u(0) \leq \infty$.
If you have $\int_1^\infty f(x)\,dx<\infty$, with $f\ge 0$ you can find a sequence $b_n\to\infty$ such that $f(b_n)\to 0$ as $n\to\infty$, since if not, then $f(x)\ge \epsilon_0>0$ for all $x$ which would give you a contradiction. It is false that $f(x)\to 0$ as $x\to\infty$. It is enough to take the graph of $f$ to be made of triangles of height 1, base $1/n^2$ and centered at each $n$. So in your case you can find $a_n\to-\infty$ and $b_n\to\infty$ such that $$((u(b_n))^2 + (u'(b_n))^2) \, w(b_n)\to0$$ and $$((u(a_n))^2 + (u'(a_n))^2) \, w(a_n)\to0.$$ Then you do the integration by parts in $[a_n,b_n]$ to get $$\int_{a_n}^{b_n} x u^2 w \, dx = \int_{a_n}^{b_n} uu'w \, dx+u^2(b_n)w(b_n)-u^2(a_n)w(a_n)$$ and then you let $n\to \infty$ noting that the integral $\int_{a_n}^{b_n} uu'w \, dx$ converges by the Lebesgue dominated convergence theorem. Similar trick for the second integration by parts probably.