Function spaces and spectra of elliptic operators on a class of hyperbolic manifolds

• Autorzy: Hans Triebel
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 35J10: Schr\"odinger operator
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 35P20: Asymptotic distribution of eigenvalues and eigenfunctions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 134
• Numer: 2
• Strony: 179-202

Daty

wydano

1999

otrzymano

1998-07-03

poprawiono

1998-11-12

Twórcy

autor

Hans Triebel

• Mathematisches Institut, Fakultät für Mathematik und Informatik, Universität Jena, D-07740 Jena, Germany,

Bibliografia

• [Ber98] G. Berger, Eigenvalue distribution of elliptic operators of second order with Neumann boundary conditions in a snowflake domain, preprint, Leipzig, 1998.
• [Dav89] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989.
• [EdE87] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Oxford Univ. Press, 1987.
• [ET96] D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Univ. Press, 1996.
• [EvH93] W. D. Evans and D. J. Harris, Fractals, trees and the Neumann Laplacian, Math. Ann. 296 (1993), 493-527.
• [Fal85] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Univ. Press, 1985.
• [Fal90] K. J. Falconer, Fractal Geometry, Wiley, Chichester, 1990.
• [Har98] D. Haroske, Some logarithmic function spaces, entropy numbers, applications to spectral theory, Dissertationes Math. 373 (1998).
• [Har99] D. Haroske, Embeddings of some weighted function spaces on $ℝ^n$; entropy and approximation numbers, preprint, Jena, 1998.
• [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.
• [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.
• [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).
• [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.
• [Mat95] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces, Cambridge Univ. Press, 1995.
• [RuS96] T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytzkij Operators, and Nonlinear Partial Differential Equations, de Gruyter, Berlin, 1996.
• [Shu92] M. A. Shubin, Spectral theory of elliptic operators on non-compact manifolds, Astérisque 207 (1992), 35-108.
• [Skr96] L. Skrzypczak, Heat semi-group and function spaces on symmetric spaces of non-compact type, Z. Anal. Anwendungen 15 (1996), 881-899.
• [Skr97] L. Skrzypczak, Besov spaces on symmetric manifolds-the atomic decomposition, Studia Math. 124 (1997), 215-238.
• [Skr98] L. Skrzypczak, Atomic decompositions on manifolds with bounded geometry, Forum Math. 10 (1998), 19-38.
• [Skr98*] L. Skrzypczak, Mapping properties of pseudodifferential operators on manifolds with bounded geometry, J. London Math. Soc., to appear.
• [Stu93] K.-T. Sturm, On the $L^p$-spectrum of uniformly elliptc operators on Riemannian manifolds, J. Funct. Anal. 118 (1993), 442-453.
• [Tay89] M. E. Taylor, $L^p$-estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793.
• [Tri83] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
• [Tri86] H. Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manfolds, Ark. Mat. 24 (1986), 299-337.
• [Tri87] H. Triebel, Characterizations of function spaces on a complete Riemannian manifold with bounded geometry, Math. Nachr. 130 (1987), 312-346.
• [Tri88] H. Triebel, On a class of weighted function spaces and related pseudodifferential operators, Studia Math. 90 (1988), 37-68.
• [Tri92] H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, 1992.
• [Tri97] H. Triebel, Fractals and Spectra, Birkhäuser, Basel, 1997.
• [Tri98] H. Triebel, Decompositions of function spaces, in: Progress in Nonlinear Differential Equations Appl. 35, Birkhäuser, 1999, 691-730.