  Let $(X,J,\omega,g)$ be a complete $n$-dimensional K\"ahler manifold. A
Theorem by Gromov \cite{G} states that the if the K\"ahler form is $d$-bounded,
then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless
$k=\frac{n}{2}$. Starting with a contact manifold $(M,\alpha)$ we show that the
same conclusion does not hold in the category of almost K\"ahler manifolds. Let
$(X,J,g)$ be a complete almost Hermitian manifold of dimension four. We prove
that the reduced $L_2$ $2^{nd}$-cohomology group decomposes as direct sum of
the closure of the invariant and anti-invariant $L_2$-cohomology. This
generalizes a decomposition theorem by Dr\v{a}ghici, Li and Zhang \cite{DLZ}
for $4$-dimensional closed almost complex manifolds to the $L_2$-setting.
