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Weak-type endpoint bounds for Riesz means
Author:
Terence Tao
Journal:
Proc. Amer. Math. Soc. 124 (1996), 2797-2805
MSC (1991):
Primary 42B15
MathSciNet review:
1327048
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Abstract: We use restriction theory to prove optimal weak-type bounds of Bochner-Riesz multipliers and Riesz means of elliptic pseudo-differential operators on compact manifolds, for .
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- J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 22 (1991), 147--187. MR 92g:42010
- 2.
- L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972): 287--299. MR 50:14052
- 3.
- M. Christ, On almost-everywhere convergence of Bochner-Riesz means in higher dimensions, Proc. Amer. Math. Soc. 95 (1985): 16--20. MR 87c:42020
- 4.
- ------, Weak type endpoint bounds for Bochner-Riesz multipliers, Revista Mat. Iberoamericana 3 (1987), 25--31. MR 90i:42024
- 5.
- ------, Weak type (1,1) bounds for rough operators, Annals of Mathematics, 128 (1988): 19--42. MR 89m:42013
- 6.
- M. Christ and C. D. Sogge, The weak-type
convergence of eigenfunction expansions for pseudo-differential operators, Invent. Math. 94 (1988): 421--453. MR 89j:35096
- 7.
- C. Feffermann, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9--36. MR 41:2468
- 8.
- ------, The multiplier problem for the ball, Ann. of Math. 94 (1971): 330--336. MR 45:5661
- 9.
- C. Feffermann and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971): 107--115. MR 44:2026
- 10.
- L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193--218. MR 58:29418
- 11.
- A. Seeger, Endpoint estimates for multiplier transformations on compact manifolds, Indiana Math. J. 40 (1991): 471-533. MR 92f:58166
- 12.
- ------, Endpoint inequalities for Bochner-Riesz multipliers in the plane, to appear, Pacific J. Math.
- 13.
- 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 91d:58244
- 14.
- C. D. Sogge, On the convergence of Riesz means on compact manifolds, Annals of Math. 126 (1987), 439--447. MR 89b:35126
- 15.
- ------, Concerning the
norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988): 123--138. MR 89d:35131
- 16.
- ------, Fourier Integrals in Classical Analysis, Cambridge Tracts in Math. # 105, Cambridge Univ. Press, 1993. MR 94c:35178
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Additional Information
Terence Tao
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email:
tao@math.princeton.edu
DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03371-0
PII:
S 0002-9939(96)03371-0
Received by editor(s):
March 14, 1995
Communicated by:
Christopher D. Sogge
Article copyright:
© Copyright 1996 American Mathematical Society
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