Krein-Feller operators on Riemannian manifolds: compactness of embedding and Hodge's theorem

Abstract

For a bounded open set Omega in a complete oriented Riemannian n-manifold and a positive finite Borel measure mu with support contained in Omega, we define an associated Krein-Feller operators (or Laplacian) Delta_mu by assuming the Poincar'e inequalities for the measure mu. We obtain sufficient conditions for the operator to have compact resolvent and in this case, we prove the Hodge theorem for functions, which states that there exists an orthonormal basis of L^2(Omega,mu) consisting of eigenfunctions of Delta_mu, the eigenspaces are finite-dimensional, and the eigenvalues of -Delta_mu are real, countable, and increasing to infinity. One of these sufficient conditions is that the lower L^infty-dimension dim_infty(mu) of mu is greater than n-2. We prove that the compactness of embedding for functions also hold for measures without compact support, provided the manifold is of bounded geometry. The main idea of our proof is to use Toponogov's and Rauch's comparison theorems to extend a classical compact embedding theorem of Maz'ja to Riemannian manifolds. For a compact Riemannian manifold, using the above results, we also obtain sufficient conditions for Hodge Laplacian on k-forms, to have compact resolvent. Our result extends the classical Hodge theorem to Krein-Feller operators. We study the condition dim_infty(mu)>n-2 for self-similar and self-conformal measures. Results in this paper extend analogous ones by Hu et al. in J. Funct. Anal., which are established for measures on R^n.