Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



New proof of the Hörmander multiplier theorem on compact manifolds without boundary

Author: Xiangjin Xu
Journal: Proc. Amer. Math. Soc. 135 (2007), 1585-1595
MSC (2000): Primary 58J40, 35P20, 35J25
Published electronically: January 9, 2007
MathSciNet review: 2276671
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: On compact manifolds $ (M, g)$ without boundary, the gradient estimates for unit band spectral projection operators $ \chi_{\lambda}$ are proved for a second order elliptic differential operator $ L$. A new proof of the Hörmander Multiplier Theorem (first proved by A. Seeger and C.D. Sogge in 1989) is given in this setting by using the gradient estimates and the Calderón-Zygmund argument.

References [Enhancements On Off] (What's this?)

  • 1. Xuan Thinh Duong, El Maati Ouhabaz, and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), no. 2, 443–485. MR 1943098, 10.1016/S0022-1236(02)00009-5
  • 2. David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
  • 3. D. Grieser, $ L^p$ bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries, Ph.D. thesis, UCLA, 1992.
  • 4. Lars Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218. MR 0609014
  • 5. Lars Hörmander, The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 274, Springer-Verlag, Berlin, 1985. Pseudodifferential operators. MR 781536
  • 6. A. Seeger and C. D. Sogge, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J. 59 (1989), no. 3, 709–736. MR 1046745, 10.1215/S0012-7094-89-05932-2
  • 7. Christopher D. Sogge, Concerning the 𝐿^{𝑝} norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123–138. MR 930395, 10.1016/0022-1236(88)90081-X
  • 8. Christopher D. Sogge, Fourier integrals in classical analysis, Cambridge Tracts in Mathematics, vol. 105, Cambridge University Press, Cambridge, 1993. MR 1205579
  • 9. Michael E. Taylor, Pseudodifferential operators, Princeton Mathematical Series, vol. 34, Princeton University Press, Princeton, N.J., 1981. MR 618463
  • 10. Xu Bin, Derivatives of the spectral function and Sobolev norms of eigenfunctions on a closed Riemannian manifold, Ann. Global Anal. Geom. 26 (2004), no. 3, 231–252. MR 2097618, 10.1023/B:AGAG.0000042902.46202.69
  • 11. Xiangjin Xu, Gradient estimates for eigenfunctions of compact manifolds with boundary and the Hörmander Multiplier Theorem (preprint).
  • 12. Xiangjin Xu, Eigenfunction Estimates on Compact Manifolds with Boundary and Hörmander Multiplier Theorem. Ph.D. Thesis, Johns Hopkins University, May, 2004.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58J40, 35P20, 35J25

Retrieve articles in all journals with MSC (2000): 58J40, 35P20, 35J25

Additional Information

Xiangjin Xu
Affiliation: Mathematical Sciences Research Institute, 17 Gauss Way, Berkeley, California 94720
Address at time of publication: Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, Virginia 22904

Keywords: Gradient estimate, eigenfunction, unit band spectral projection operator, H\"ormander Multiplier Theorem
Received by editor(s): September 15, 2005
Received by editor(s) in revised form: February 28, 2006
Published electronically: January 9, 2007
Communicated by: Andreas Seeger
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.