Abstract
We consider weighted function spaces of Sobolev-Besov type and Schrödinger 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bergh, J., Löfström, J.: Interpolation Theory. An Introduction. Springer, Berlin (1976)
Birman, M.S., Solomyak, M.Z.: Quantitative analysis in Sobolev embedding theorems and applications to spectral theory. AMS Transl. 114, 1–132 (1980)
Birman, M.S., Solomyak, M.Z.: Estimates for the numbers negative eigenvalues of the Schrödinger operator and its generalizations. Adv. Sov. Math. 7, 1–55 (1991)
Bourdaud, G.: Ondelettes et espaces de Besov. Rev. Math. Iberoam. 11, 477–512 (1995)
Buy, H.-Q., Paluszyński, M., Taibleson, M.H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Stud. Math. 119, 219–246 (1996)
Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge University Press, Cambridge (1990)
Carl, B., Triebel, H.: Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces. Math. Ann. 251, 129–133 (1980)
Chavel, I.: Riemannian Geometry—A Modern Introduction. Cambridge University Press, Cambridge (1993)
Cheeger, J., Gromov, M., Taylor, M.: Finite Propagation speed, Kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)
Donnelly, H.: On the essential spectra of a complete Riemannian manifold. Topology 20, 1–14 (1981)
Edmunds, D.E., Evans, W.D.: Spectral Theory and Differential Operators. Clarendon, Oxford (1987)
Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996)
Haroske, D., Triebel, H.: Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudo differential operators II. Math. Nachr. 168, 109–137 (1994)
Haroske, D., Triebel, H.: Wavelet bases and entropy numbers in weighted function spaces. Math. Nachr. 278, 108–132 (2005)
Kühn, Th.: Entropy numbers in weighted function Besov spaces. The case of Intermediate weights. Proc. Steklov Math. Inst. 255, 159–168 (2006)
Kühn, Th., Leopold, H.-G., Sickel, W., Skrzypczak, L.: Entropy Numbers of Embeddings of Weighted Besov Spaces. Jenaer Schriften zur Mathematik und Informatik Math/Inf/13/03
Kühn, Th., Leopold, H.-G., Sickel, W., Skrzypczak, L.: Entropy numbers of embeddings of weighted Besov spaces. Constr. Approx. 23, 61–77 (2006)
Kühn, Th., Leopold, H.-G., Sickel, W., Skrzypczak, L.: Entropy numbers of embeddings of weighted Besov spaces, II. Proc. Edinb. Math. Soc. (2) 49, 331–359 (2006)
Kühn, Th., Leopold, H.-G., Sickel, W., Skrzypczak, L.: Entropy numbers of embeddings of weighted Besov spaces III: weights of logarithmic type. Math. Z. 255, 1–15 (2007)
Kondrat’ev, V., Shubin, M.: Discreteness of the spectrum for the Schrödinger operators on manifolds with bounded geometry. Oper. Theory Adv. Appl. 100, 185–226 (1999)
Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)
Shubin, M.A.: Spectral theory of elliptic operators on non-compact manifolds. Astérisque 207, 37–108 (1992)
Shubin, M.A.: Spectral theory the Schrödinger operators on non-compact manifolds: qualitative results. In Spectral Theory and Geometry, pp. 226–283. Cambridge University Press, Cambridge (1999)
Skrzypczak, L.: Atomic decompositions on manifolds with bounded geometry. Forum Math. 10, 19–38 (1998)
Taylor, M.: L p Estimates on functions of the Laplace operator. Duke Math. J. 58, 773–793 (1989)
Triebel, H.: Function spaces and spectra of elliptic operators on a class of hyperbolic manifolds. Stud. Math. 134, 179–202 (1999)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)
Triebel, H.: Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds. Arkiv Mat. 24, 300–337 (1986)
Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)
Triebel, H.: The Structure of Functions. Birkhäuser, Basel (2001)
Yau, S.T.: Some function-theoretical properties of complete Riemannian manifolds and their applications to geometry. Indiana Math. J. 25, 659–670 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Stephan Dahlke.
Rights and permissions
About this article
Cite this article
Skrzypczak, L. Wavelet Frames, Sobolev Embeddings and Negative Spectrum of Schrödinger Operators on Manifolds with Bounded Geometry. J Fourier Anal Appl 14, 415–442 (2008). https://doi.org/10.1007/s00041-008-9016-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00041-008-9016-2
Keywords
- Weighted Sobolev and Besov spaces
- Wavelet frames
- Compact embeddings
- Entropy numbers
- Schrödinger operators
- Negative spectrum