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On the maximal Bochner-Riesz conjecture in the plane for $p<2$

Author: Terence Tao
Journal: Trans. Amer. Math. Soc. 354 (2002), 1947-1959
MSC (2000): Primary 42B15
Published electronically: January 7, 2002
MathSciNet review: 1881025
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Abstract: We give a new estimate on the maximal Bochner-Riesz operator in the plane, for $p<2$; as a corollary we obtain an almost everywhere convergence result for certain Bochner-Riesz means. This work was inspired by discussions with Michael Christ and Chris Sogge.

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  • 1. J. Bourgain, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. and Funct. Anal. 22 (1991), 147-187. MR 92g:42010
  • 2. J. Bourgain, A remark on Schrodinger operators, Israel J. Math. 77 (1992), 1-16. MR 93k:35071
  • 3. J. Bourgain, Estimates for cone multipliers, Operator Theory: Advances and Applications, 77 (1995), 41-60. MR 96m:42022
  • 4. J. Bourgain, Some new estimates on oscillatory integrals, Essays in Fourier Analysis in Honor of E. M. Stein, Princeton University Press (1995), 83-112. MR 96c:42028
  • 5. J. Bourgain, Refinements of Strichartz' Inequality and Applications to 2D-NLS with Critical Nonlinearity, Internat. Math. Res. Notes 1998, no. 5, 253-283. MR 99f:35184
  • 6. A. Carbery, The boundedness of the maximal Bochner-Riesz operator on $L^4(\textbf{R}^2)$, Duke Math. J. 50 (1983), 409-416. MR 84m:42025
  • 7. A. Carbery, Variants of the Caldersn-Zygmund theory for $L\sp p$-spaces, Rev. Mat. Iberoamericana 2 (1986), 381-396. MR 89f:42011
  • 8. A. Carbery; J. L. Rubio de Francia, L. Vega, Almost everywhere summability of Fourier integrals. J. London Math. Soc. 38 (1988), 513-524. MR 90e:42033
  • 9. L. Carleson and P. Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972): 287-299. MR 50:14052
  • 10. L. Chen, D. Fan, The convergence of the Bochner-Riesz means at the critical index, Proc. Amer. Math. Soc. 124 (1996). MR 96k:42014
  • 11. M. Christ, On the regularity of inverses of singular integral operators, Duke Math. J., 57 (1988): 459-484. MR 90c:42022
  • 12. C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. MR 41:2468
  • 13. C. Fefferman, The multiplier problem for the ball, Ann. of Math. 94 (1971): 330-336. MR 45:5661
  • 14. I. L. Hwang, The $L^2$-boundedness of pseudodifferential operators, Trans. Amer. Math. Soc. 302 (1987), 55-76. MR 88e:47096
  • 15. Y. Kanjin, Convergence almost everywhere of Bochner-Riesz means for radial functions, Ann. Sci. Kanazawa Univ. 25 (1988): 11-15. MR 90b:42034
  • 16. M. Kojima, On the almost everywhere convergence of Bochner-Riesz means of multiple Fourier integrals for radial functions., Nihonkai Math. J. 3 (1992), 9-21. MR 93f:42027
  • 17. S. Z. Lu, Decomposition of kernel and maximal generalized Bochner-Riesz means, Chinese Quart. J. Math. 4 (1989): 16-23. MR 90k:42016
  • 18. A. Moyua, A. Vargas, L. Vega, Schrödinger Maximal Function and Restriction Properties of the Fourier Transform, International Math. Research Notices 1996, no. 16, 793-815. MR 97k:42042
  • 19. A. Moyua, A. Vargas, L. Vega, Restriction theorems and maximal operators related to oscillatory integrals in $\textbf{R}^3$, Duke Math. J. 96 (1999), 547-574. MR 2000b:42017
  • 20. E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956): 482-492. MR 18:575d
  • 21. E. M. Stein, On limits of sequences of operators, Ann. of Math. 74 (1961): 140-170. MR 23:A2695
  • 22. E. M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978), Part 1, pp. 3-20. MR 80m:42027
  • 23. E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. MR 46:4102
  • 24. E. M. Stein, Harmonic Analysis, Princeton University Press, 1993. MR 95c:42002
  • 25. E. M. Stein, M. H. Taibleson, and G. Weiss, Weak-type estimates for maximal operators on certain $H^p$ spaces, Rend. Circ. Mat. Palermo suppl. 1 (1981): 81-97. MR 83c:42017
  • 26. T. Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math J. 96 (1999), 363-375. MR 2000a:42023
  • 27. T. Tao, The weak-type endpoint Bochner-Riesz conjecture and related topics, Indiana Univ. Math. J. 47 (1998), 1097-1124. MR 2000a:42024
  • 28. T. Tao, A. Vargas, L. Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967-1000. MR 99f:42026

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Additional Information

Terence Tao
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90024

Received by editor(s): January 30, 1998
Published electronically: January 7, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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