# 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?