Let $X$ be a complex manifold. Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact complex manifold $X$, $H^{p,q}(X) \cong H^{p,q}_{(2)}(X)$ by the Hodge isomorphism. On a noncompact manifold, this is no longer holds in general. See https://mathoverflow.net/a/362016/124749 .
My question is : Is there a general principle to compare those 2 type cohomologies on a noncompact manifold. If not, what will happen to that on Stein manifolds?
