Hodge Theorem for Krein-Feller operators on compact Riemannian manifolds

Abstract

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.