Orthogonal ("Hodge") Decomposition of $L^2(\Omega)$ with $\Omega$ being unbounded

In Mathematical Analysis and Numerical Methods for Science and Technology: Volume 3 Spectral Theory and Applications it is stated on page 313 that for a regular and bounded set $$\Omega\subset\mathbb{R}^n$$ several decomposition for $$L^2(\Omega)^n$$ apply, such as $$L^2(\Omega)^n=\rm{grad}\,H_0^1(\Omega)\oplus H(\rm{div}\,0,\Omega).$$ A footnote present on the same page states that in the case where $$\Omega$$ is the complement of a bounded set, there exist analogous equalities which result by replacing the Sobolev spaces with certain Beppo Levi spaces. Is someone aware of a resource which states this result in a more explicit way?

These spaces are also called Deny-Lions Spaces in that case, but that book was co-authored by Lions so he did not name it that way.

An example of such use is here.

I think the main reference is Amrouche Girault and Giroire, and if you look at the google scholar citation of that paper you will find a number of articles where this approach is developed.

The general idea is to map back from outside to inside, and then apply the results of Girault and Raviart for example (or others).