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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
Posted:
January 9, 2007
MathSciNet review:
2276671
Full-text PDF Free Access
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Abstract: On compact manifolds  without boundary, the gradient estimates for unit band spectral projection operators  are proved for a second order elliptic differential operator  . 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.
- 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
(2003k:43012), http://dx.doi.org/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
(2001k:35004)
- 3.
D. Grieser,
 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
(58 #29418)
- 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
(87d:35002a)
- 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
(91d:58244), http://dx.doi.org/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
(89d:35131), http://dx.doi.org/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
(94c:35178)
- 9.
Michael
E. Taylor, Pseudodifferential operators, Princeton
Mathematical Series, vol. 34, Princeton University Press, Princeton,
N.J., 1981. MR
618463 (82i:35172)
- 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
(2005h:58051), http://dx.doi.org/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.
- 1.
- Xuan Thinh Duong, El Maati Ouhabaz, and Adam Sikora, Plancherel-type estimates and sharp spectral multipliers. J. funct. Anal. 196 (2002) 443-485.MR 1943098 (2003k:43012)
- 2.
- D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, 2001. MR 1814364 (2001k:35004)
- 3.
- D. Grieser, bounds for eigenfunctions and spectral projections of the Laplacian near concave boundaries, Ph.D. thesis, UCLA, 1992.
- 4.
- L. Hörmander, The spectral function of an elliptic operator, Acta Math. 88 (1968), 341-370.MR 0609014 (58:29418)
- 5.
- L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag, 1985.MR 0781536 (87d:35002a)
- 6.
- A. Seeger and C. D. Sogge, On the boundedness of functions of pseudo-differential operators on compact manifolds. Duke Math. J. 59 (1989), 709-736.MR 1046745 (91d:58244)
- 7.
- C. D. Sogge, Concerning the
 norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), no. 1, 123-134.MR 0930395 (89d:35131)
- 8.
- C. D. Sogge, Fourier integrals in classical analysis. Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993.MR 1205579 (94c:35178)
- 9.
- M. Taylor, Pseudo-differential Operators. Princeton Univ. Press, Princeton N.J., 1981.MR 0618463 (82i:35172)
- 10.
- Bin Xu, Derivatives of the Spectral Function and Sobolev Norms of Eigenfunctions on a Closed Riemannian Manifold. Annals of Global Analysis and Geometry 26 (2004), 231-252. MR 2097618 (2005h:58051)
- 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.
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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
Email:
xiangjxu@msri.org, xx8n@virginia.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08687-X
PII:
S 0002-9939(07)08687-X
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
Posted:
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.
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