The paper deals with quarkonial decompositions and entropy numbers in weighted function spaces on hyperbolic manifolds. We use these results to develop a spectral theory of related Schrödinger operators in these hyperbolic worlds.
@article{bwmeta1.element.bwnjournal-article-smv134i2p179bwm,
author = {Hans Triebel},
title = {Function spaces and spectra of elliptic operators on a class of hyperbolic manifolds},
journal = {Studia Mathematica},
volume = {133},
year = {1999},
pages = {179-202},
zbl = {0935.46033},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv134i2p179bwm}
}
Triebel, Hans. Function spaces and spectra of elliptic operators on a class of hyperbolic manifolds. Studia Mathematica, Tome 133 (1999) pp. 179-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv134i2p179bwm/
[00000] [Ber98] G. Berger, Eigenvalue distribution of elliptic operators of second order with Neumann boundary conditions in a snowflake domain, preprint, Leipzig, 1998.
[00001] [Dav89] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989.
[00002] [EdE87] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Univ. Press, 1987.
[00003] [ET96] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, 1996. | Zbl 0865.46020
[00004] [EvH93] W. D. Evans and D. J. Harris, Fractals, trees and the Neumann Laplacian, Math. Ann. 296 (1993), 493-527. | Zbl 0799.46041
[00005] [Fal85] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, 1985.
[00006] [Fal90] K. J. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
[00007] [Har98] D. Haroske, Some logarithmic function spaces, entropy numbers, applications to spectral theory, Dissertationes Math. 373 (1998). | Zbl 0906.46027
[00008] [Har99] D. Haroske, Embeddings of some weighted function spaces on ; entropy and approximation numbers, preprint, Jena, 1998.
[00009] [HaT94] D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators I, Math. Nachr. 167 (1994), 131-156. | Zbl 0829.46019
[00010] [HaT94*] D. Haroske and H. Triebel, Entropy numbers in weighted function spaces and eigenvalue distributions of some degenerate pseudodifferential operators II, ibid. 168 (1994), 109-137. | Zbl 0829.46020
[00011] [HeL97] C. Q. He and M. L. Lapidus, Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function, Mem. Amer. Math. Soc. 608 (1997).
[00012] [Lap91] M. L. Lapidus, Fractal drums, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Amer. Math. Soc. 325 (1991), 465-529. | Zbl 0741.35048
[00013] [Mat95] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 1995.
[00014] [RuS96] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytzkij Operators, and Nonlinear Partial Differential Equations, de Gruyter, Berlin, 1996.
[00015] [Shu92] M. A. Shubin, Spectral theory of elliptic operators on non-compact manifolds, Astérisque 207 (1992), 35-108. | Zbl 0793.58039
[00016] [Skr96] L. Skrzypczak, Heat semi-group and function spaces on symmetric spaces of non-compact type, Z. Anal. Anwendungen 15 (1996), 881-899. | Zbl 0868.46025
[00017] [Skr97] L. Skrzypczak, Besov spaces on symmetric manifolds-the atomic decomposition, Studia Math. 124 (1997), 215-238. | Zbl 0896.46024
[00018] [Skr98] L. Skrzypczak, Atomic decompositions on manifolds with bounded geometry, Forum Math. 10 (1998), 19-38. | Zbl 0907.46031
[00019] [Skr98*] L. Skrzypczak, Mapping properties of pseudodifferential operators on manifolds with bounded geometry, J. London Math. Soc., to appear.
[00020] [Stu93] K.-T. Sturm, On the -spectrum of uniformly elliptc operators on Riemannian manifolds, J. Funct. Anal. 118 (1993), 442-453.
[00021] [Tay89] M. E. Taylor, -estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793.
[00022] [Tri83] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
[00023] [Tri86] H. Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manfolds, Ark. Mat. 24 (1986), 299-337. | Zbl 0664.46026
[00024] [Tri87] H. Triebel, Characterizations of function spaces on a complete Riemannian manifold with bounded geometry, Math. Nachr. 130 (1987), 312-346.
[00025] [Tri88] H. Triebel, On a class of weighted function spaces and related pseudodifferential operators, Studia Math. 90 (1988), 37-68.
[00026] [Tri92] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
[00027] [Tri97] H. Triebel, Fractals and Spectra, Birkhäuser, Basel, 1997.
[00028] [Tri98] H. Triebel, Decompositions of function spaces, in: Progress in Nonlinear Differential Equations Appl. 35, Birkhäuser, 1999, 691-730.