I just want to check something. I want to use the energy estimates on the real line for an elliptic operator $L$ acting on $L^2(\mathbb{R})$. (The energy estimates are related to the Lax-Mi
https://www.maths.ed.ac.uk/~aram/weak.pdf (Slide 17)
The equation is given as $Lu = 0$, and $u$ has boundary conditions $u \longrightarrow 0$ as $x \longrightarrow \infty$. Furthermore, the domain of the operator is $L:D(L) \rightarrow X$ where $D(L) = H_0^1(\mathbb{R}) \cap H^2(\mathbb{R})$.
The reason why I am apprehensive about using this is largely from the use of the Poincare inequality that doesn't work on $\mathbb{R}$. My understanding however is that I should be able to use the energy estimates though.