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New proof of the Hörmander multiplier theorem on compact manifolds without boundary
Author(s):
Xiangjin
Xu
Journal:
Proc. Amer. Math. Soc.
135
(2007),
1585-1595.
MSC (2000):
Primary 58J40, 35P20, 35J25
Posted:
January 9, 2007
MathSciNet review:
2276671
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Additional information
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.
References:
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- 1.
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- 2.
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- 3.
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 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)
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 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.
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- M. Taylor, Pseudo-differential Operators. Princeton Univ. Press, Princeton N.J., 1981.MR 0618463 (82i:35172)
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- 11.
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- 12.
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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:
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
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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