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Maximal averages along a planar vector field depending on one variable


Author: Michael Bateman
Journal: Trans. Amer. Math. Soc. 365 (2013), 4063-4079
MSC (2010): Primary 42B25; Secondary 42B20
DOI: https://doi.org/10.1090/S0002-9947-2013-05673-5
Published electronically: March 12, 2013
MathSciNet review: 3055689
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove (essentially) sharp $ L^2$ estimates for a restricted maximal operator associated to a planar vector field that depends only on the horizontal variable. The proof combines an understanding of such vector fields from earlier work of the author with a result of Nets Katz on directional maximal operators.


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

  • 1. Bateman, Michael. $ L^p$ estimates for maximal averages along one-variable vector fields in $ {\mathbf R} ^2$, Proc. Amer. Math. Soc. $ \mathbf {137}$, (2009), 955-963. MR 2457435 (2010e:42023)
  • 2. Duoandikoetxea, Javier. Fourier Analysis, 2001, AMS Graduate Studies in Mathematics, vol. 29. MR 1800316 (2001k:42001)
  • 3. Fefferman, C. Pointwise convergence of Fourier series, Annals of Mathematics. Second Series 98 (3): 551-571. MR 0340926 (49:5676)
  • 4. Katz, N.H. Maximal operators over arbitrary sets of directions, Duke Math. J. vol. 97, no. 1 (1999), 67-79. MR 1681088 (2000a:42036)
  • 5. Katz, N.H. Remarks on maximal operators over arbitrary sets of directions, Bull. London Math. Soc. vol. 31 (1999), no. 6, pages 700-710. MR 1711029 (2001g:42041)
  • 6. Lacey, Michael, and Li, Xiaochun. Maximal Theorems for the Directional Hilbert Transform on the Plane, Trans. Amer. Math. Soc. 358 (2006), 4099-4117. MR 2219012 (2006k:42018)
  • 7. Lacey, Michael, and Li, Xiaochun. On a Conjecture of EM Stein on the Hilbert Transform on Vector Fields, Memoirs of the AMS 205 (2010), no. 965. MR 2654385 (2011c:42019)
  • 8. Lacey, Michael, and Li, Xiaochun. On a Lipschitz Variant of the Kakeya Maximal Function. Available at http://arxiv.org/abs/math/0601213

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Additional Information

Michael Bateman
Affiliation: Department of Mathematics, University of California, Los Angeles, Box 951555, Los Angeles, California 90095-1555
Address at time of publication: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
Email: bateman@math.ucla.edu, m.bateman@dpmms.com.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2013-05673-5
Received by editor(s): June 20, 2011
Received by editor(s) in revised form: July 15, 2011
Published electronically: March 12, 2013
Additional Notes: This work was supported by NSF grant DMS-0902490
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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