Wavelet Frames, Sobolev Embeddings and Negative Spectrum of Schrödinger Operators on Manifolds with Bounded Geometry

Abstract

We consider weighted function spaces of Sobolev-Besov type and Schrö-dinger type operators on noncompact Riemannian manifolds with bounded geometry. First we give characterization of the spaces in terms of wavelet frames. Then we describe the necessary and sufficient conditions for the compactness of Sobolev embeddings between the spaces. An asymptotic behavior of the corresponding entropy numbers is calculated. At the end we use the asymptotic behavior to estimate the number of negative eigenvalues of the Schrödinger type operators.

Parseval wavelet frames on Riemannian manifold

“…More precisely, it is assumed that M is a connected complete Riemannian manifold with positive injectivity radius and bounded geometry. The theory of Triebel-Lizorkin and Besov spaces on such manifolds was further developed by Triebel [39], Skrzypczak [35], and Große and Schneider [21]. Our boundedness result is an extension of analogous result for Sobolev spaces shown in [3].…”

Smooth Orthogonal Projections on Riemannian Manifold

“…We show a decomposition of Sobolev spaces on manifolds extending results of Ciesielski and Figiel [6,7,8] for compact manifolds and the first two authors [4] for the sphere. In addition, Triebel [21,22] has extended the theory of Triebel-Lizorkin and Besov spaces on complete Riemannian manifolds with bounded geometry, see also [20] and [23,Ch. 7].…”

Solutions of the Equation of a Spinorial Yamabe-type Problem on Manifolds of Bounded Geometry

“…In [15], continuity and compactness for weighted Sobolev embeddings were studied for Sobolev spaces of real valued functions. We will need the same result for Sobolev spaces of bundles, especially the compactness of the embedding H q 1 (M, S) ֒→ ρL p (M, S) on spinors for q and p conjugate, ρ a radial weight (see A.1) and 2 ≤ p < p crit = 2n n−1 .…”

“…We will need the same result for Sobolev spaces of bundles, especially the compactness of the embedding H q 1 (M, S) ֒→ ρL p (M, S) on spinors for q and p conjugate, ρ a radial weight (see A.1) and 2 ≤ p < p crit = 2n n−1 . In this Appendix, we want to use the result from Skrzypczak [15] to obtain the following theorem (or more generally a bundle version of Theorem 24): Theorem 21. Let (M, g) be an n-dimensional manifold with an hermitian vector bundle E of bounded geometry (e.g.…”