  For an open set in a compact smooth oriented Riemannian n-manifold and a
positive finite Borel measure with support contained in the closure of the open
set, we define an associated Krein-Feller operator on k-forms by assuming the
Poincare inequality. Krein-Feller operators on Euclidean space have been
studied extensively in fractal geometry. Using results established by the
authors, we obtain sufficient conditions for Krein-Feller operator to have
compact resolvent. Under these conditions, we prove the Hodge theorem for
forms, which states that there exists an orthonormal basis of L2 consisting of
eigenforms of Krein-Feller operator, the eigenspaces are finite-dimensional,
and the eigenvalues of Krein-Feller operator are real, countable, and
increasing to infinity. One of these sufficient conditions is that the
dimension of the measure is greater than n-2. Our result extends the classical
Hodge theorem to Krein-Feller operators.
