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.

Figures (4)

• Figure 1: The concentric balls $B(z,\rho)$ and $B(z,\rho+2\delta)$ inside the concentric ball $B(z,\epsilon)=\varphi(B^M(p,\epsilon))$ (not shown). The shaded part is the set $\widetilde{V}_{\delta}(x)$.
• Figure 2: Figure (a) shows the concentric balls $B^M(p,\rho)$ and $B^M(p,\rho+2\delta)$ in the domain $B^M(p,\epsilon)$ of $\varphi$. The shaded part is the set $V_{\delta}(\varphi^{-1}(x))$. Figure (b) shows the set $M^{\underline{\kappa}(W)}$ (or $M^{\overline{\kappa}(W)}$), where $\underline{\kappa}(W)<0$ (or $\overline{\kappa}(W)<0$). Figure (c) shows the set $M^{\underline{\kappa}_W}$ (or $M^{\overline{\kappa}(W)}$), where $\underline{\kappa}(W)=0$ (or $\overline{\kappa}(W)=0$). Figure (d) shows the set $M^{\underline{\kappa}(W)}$ (or $M^{\overline{\kappa}(W)}$), where $\underline{\kappa}(W)>0$ (or $\overline{\kappa}(W)>0$). The shaded part of each of the figures (b), (c) and (d) is an Aleksandrow triangle in $M^{\underline{\kappa}(W)}$ (or $M^{\overline{\kappa}(W)}$).
• Figure 3: A possible arrangement of some elements of the sets $\mathcal{L}_k$, $k=1,\ldots,7$. Note that in this example a geodesic ball can intersect at most six other geodesic balls. Step II.1. We choose a geodesic ball that intersects with six geodesic balls and name it $B_{1,1}$. Then, we choose another geodesic ball that does not intersect $B_{1,1}$ but intersects the other six geodesic balls, and denote it by $B_{1,2}$. Step II.2. From the geodesic balls intersecting $B_{1,1}$, we choose one and name it $B_{2,1}^1$. Then, from the geodesic balls intersecting $B_{1,2}$ but not $B_{1,1}$, we choose one and name it $B_{2,2}^1$. Step II.2a. We choose a geodesic ball that intersects five other geodesic balls and name it $B_{2,1}^2$. Step II.3. From the geodesic balls intersecting $B_{1,1}$, we choose one and name it $B_{3,1}^1$. Then, from the geodesic balls intersecting $B_{1,2}$ but not $B_{1,1}$, we choose one and name it $B_{3,2}^1$. Again, from the geodesic balls intersecting $B_{2,1}^2$ but not $B_{1,1}\cup B_{1,2}$, we choose another and name it $B_{3,1}^2$. Since none of the remaining geodesic balls intersect four other geodesic balls, we stop this step. Step II.4. From the geodesic balls intersecting $B_{1,1}$, we choose one and denote it by $B_{4,1}^1$. Since $B_{4,1}^1\cap B_{1,2}\neq\emptyset$, there is no need to choose another geodesic ball from those intersecting $B_{1,2}$. Then, from the geodesic balls intersecting with $B_{2,1}^2$ but not $B_{1,1}\cup B_{1,2}$, we choose one and denote it by $B_{4,1}^2$. Since none of remaining geodesic balls intersect three other geodesic balls, we stop this step. Step II.5. From the geodesic balls intersecting $B_{1,1}$, we choose one and denote it by $B_{5,1}^1$. Then, from the geodesic balls intersecting $B_{1,2}\backslash B_{1,1}$, we choose one and name it $B_{5,2}^1$. Since geodesic balls $B_{5,2}^1\cap B_{2,1}^2\neq \emptyset$, there is no need to choose another geodesic ball from those intersecting $B_{2,1}^2$. Since none of the remaining geodesic balls intersect with two other geodesic balls, we stop this step. Step II.6. From the geodesic balls intersecting $B_{1,1}$, we choose one and name it $B_{6,1}^1$. Since $B_{5,2}^1$, $B_{1,2}$, and $B_{2,1}^2$ all intersect each other, there is no need to choose another geodesic ball from those intersecting $B_{1,2}$ and $B_{2,1}^2$. Since none of the remaining geodesic balls intersect only one geodesic ball, we stop this step. Step II.7. Only one geodesic ball remains, which we call $B_{7,1}^1$.
• Figure 4: (a) shows the set $\varphi(B^M(p,\rho+2\delta))$ in $\mathbb{R}^3$. (b) shows, for a fixed $k$, a 2-dimensional cross-section of the 3-dimensional smooth manifold $M$. The shaded part of (b) is $R_{\delta}^M$.

Theorems & Definitions (94)

• Theorem 2.1
• Theorem 2.2
• Definition 2.1
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Definition 2.2
• Theorem 2.8