  We develop a Fredholm Theory for the Hodge Laplacian in weighted spaces on
ALG$^*$ manifolds in dimension four. We then give several applications of this
theory. First, we show the existence of harmonic functions with prescribed
asymptotics at infinity. A corollary of this is a non-existence result for
ALG$^*$ manifolds with non-negative Ricci curvature having group $\Gamma =
\{e\}$ at infinity. Next, we prove a Hodge decomposition for the first de Rham
cohomology group of an ALG$^*$ manifold. A corollary of this is vanishing of
the first betti number for any ALG$^*$ manifold with non-negative Ricci
curvature. Another application of our analysis is to determine the optimal
order of ALG$^*$ gravitational instantons.
