So, I was looking at the definition for the $H_0^1(\Omega)$ space, and I was wondering that if $\Omega = \mathbb{R}$.

This is really to embed the boundary conditions of my operator, $\mathcal{L}$ which is posed on the real line, but also has conditions such that for $\frac{du}{dt} = \mathcal{L}u$, $u \longrightarrow 0$ as $x\longrightarrow \pm \infty$.

For the space $H_0^1(\Omega)$, the norm is given by $||\frac{du}{dx} ||_{L^2}$, but this norm is motivated using Poincare's inequality that I am not sure can be used on the entire real line? On wikipedia for Poincare's inequality, it says that it must be bounded in at least one direction.