Semi-classical functional calculus on manifolds with ends and weighted L p estimates

Jean-Marc Bouclet[1]

  • [1] Université Paul Sabatier - IMT UMR CNRS 5219 31062 Toulouse Cedex 9 (France)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 3, page 1181-1223
  • ISSN: 0373-0956

Abstract

top
For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related L p boundedness properties of these operators and show in particular that, although they are not bounded on L p in general, they are always bounded on suitable weighted L p spaces.

How to cite

top

Bouclet, Jean-Marc. "Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates." Annales de l’institut Fourier 61.3 (2011): 1181-1223. <http://eudml.org/doc/219846>.

@article{Bouclet2011,
abstract = {For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related $L^p$ boundedness properties of these operators and show in particular that, although they are not bounded on $L^p$ in general, they are always bounded on suitable weighted $L^p$ spaces.},
affiliation = {Université Paul Sabatier - IMT UMR CNRS 5219 31062 Toulouse Cedex 9 (France)},
author = {Bouclet, Jean-Marc},
journal = {Annales de l’institut Fourier},
keywords = {Manifold with ends; $L^p$ estimates; $h$-pseudodifferential operators; manifold with ends; estimates; -pseudodifferential operators},
language = {eng},
number = {3},
pages = {1181-1223},
publisher = {Association des Annales de l’institut Fourier},
title = {Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates},
url = {http://eudml.org/doc/219846},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Bouclet, Jean-Marc
TI - Semi-classical functional calculus on manifolds with ends and weighted $L^p$ estimates
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 3
SP - 1181
EP - 1223
AB - For a class of non compact Riemannian manifolds with ends, we give semi-classical expansions of bounded functions of the Laplacian. We then study related $L^p$ boundedness properties of these operators and show in particular that, although they are not bounded on $L^p$ in general, they are always bounded on suitable weighted $L^p$ spaces.
LA - eng
KW - Manifold with ends; $L^p$ estimates; $h$-pseudodifferential operators; manifold with ends; estimates; -pseudodifferential operators
UR - http://eudml.org/doc/219846
ER -

References

top
  1. B. Ammann, R. Lauter, V. Nistor, A. Vasy, Complex powers and non compact manifolds, Comm. PDE 29 (2004), 671-705 Zbl1071.58022MR2059145
  2. R. Beals, Characterization of pseudo-differential operators and applications, Duke Math. J. 44 (1977), 45-57 Zbl0353.35088MR435933
  3. J. M. Bony, Caractérisation des opérateurs pseudo-différentiels, (1996-1997), Séminaire X-EDP, exp. XXIII Zbl1061.35531MR1482829
  4. J. M. Bouclet, Strichartz estimates on asymptotically hyperbolic manifolds Zbl1230.35027MR2783305
  5. J. M. Bouclet, Littlewood-Paley decompositions on manifolds with ends, Bulletin de la SMF 138, fascicule 1 (2010), 1-37 Zbl1198.42013MR2638890
  6. J. M. Bouclet, N. Tzvetkov, Strichartz estimates for long range perturbations, Amer. J. Math. 129 (2007), 1565-1609 Zbl1154.35077MR2369889
  7. N. Burq, P. Gérard, N. Tzvetkov, Strichartz inequalities and the non linear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), 569-605 Zbl1067.58027MR2058384
  8. J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds, J. Diff. Geom. 17 (1982), 15-53 Zbl0493.53035MR658471
  9. J. L. Clerc, E. M. Stein, L p -multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911-3912 Zbl0296.43004MR367561
  10. E. B. Davies, Spectral theory and differential operators, (1995), Cambridge University Press Zbl0893.47004MR1349825
  11. M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, 268 (1999), Cambridge University Press Zbl0926.35002MR1735654
  12. G. Grubb, Functionnal calculus of pseudo-differential boundary problems, 65 (1986), Birkhäuser, Boston Zbl0622.35001
  13. A. Hassel, T. Tao, J. Wunsch, A Strichartz inequality for the Schrödinger equation on non-trapping asymptotically conic manifolds, Comm. PDE 30 (2004), 157-205 Zbl1068.35119MR2131050
  14. B. Helffer, D. Robert, Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles, J. Funct. Analysis 53 (1983), 246-268 Zbl0524.35103MR724029
  15. Y. A. Kordyukov, L p estimates for functions of elliptic operators on manifolds of bounded geometry, Russian J. Math. Phys. 7 (2000), 216-229 Zbl1065.58021MR1836640
  16. R. B. Melrose, Geometric scattering theory, (1995), Cambridge Univ. Press Zbl0849.58071MR1350074
  17. D. Robert, Autour de l’approximation semi-classique, 68 (1987), Birkhaüser Zbl0621.35001MR897108
  18. B. W. Schulze, Pseudo-differential operators on manifolds with singularities, (1991), North-Holland, Amsterdam Zbl0747.58003MR1142574
  19. R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp. in Pure Math. 10 (1967), 288-307 Zbl0159.15504MR237943
  20. R. T. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969), 889-920 Zbl0191.11801MR265764
  21. E.M. Stein, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Zbl0207.13501MR290095
  22. M. Taylor, L p estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793 Zbl0691.58043MR1016445
  23. M. Taylor, Partial Differential Equations II, Linear Equations, 116 (1996), Springer Zbl0869.35003MR1395149
  24. M. Taylor, Partial Differential Equations III, Nonlinear Equations, 117 (1996), Springer Zbl1206.35004MR1477408

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.