  Let $\Omega$ be a domain with noncompact boundary. It is known that the
Helmholtz decomposition is not always valid in $L^p(\Omega)$ except for the
energy space $L^2 (\Omega)$. In this paper we consider a typical unbounded
domain whose boundary is given as a Lipschitz graph, and show that the
Helmholtz decomposition holds in certain anisotropic spaces which include some
infinite energy vector fields.
