For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x \rangle = (1+|x|^2)^{\frac{1}{2}}$

Let $ S = \{x\in \mathbb{R}^3: a<x_1<b \}$. Note the following easy Poincaré inequality first:
$\| \partial_1 u \|_{L^2(S)} > \frac{\sqrt{2}}{b-a} \|u\|_{L^2(S)}$ for all  $ u \in C^{\infty}_c(S)$. 

My question is whether I can write the above inequality globally using weights. Let $-1<\delta <1 $ and $\delta $ is not equal to zero.
Prove that there exist a constant C such that:

$\| \partial_1 u \|_{L^2_{\delta+1}} > C \|u\|_{L^2_{\delta-1}}$ for all $ u \in C^{\infty}_c (\mathbb{R}^3)$ ?

Also prove that the following stronger inequality also holds at least for values of $\delta$ sufficiently close to $-1$:
$ 4 \| \partial_1 u \|^2_{L^2_{\delta+1}} - (\delta+1)^2 \int x_1^2 \langle x \rangle^{\delta-3}u^2 \geq C \|u\|^2_{L^2_{\delta-1}} $

Thanks,

Comment: I have a way of proving both these hypothesis in a way different to the solution below. I will post the solution later if there was any interest.