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

Let $ S = \{x\in \mathbb{R}^3: a<x_1<b \}$Note the following easy poincare 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. More specifically does 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)$ ?

Thanks,