Maximal theorems for the directional Hilbert transform on the plane
Authors:
Michael T. Lacey and Xiaochun Li
Journal:
Trans. Amer. Math. Soc. 358 (2006), 40994117
MSC (2000):
Primary 42B20, 42B25; Secondary 42B05
Published electronically:
March 24, 2006
MathSciNet review:
2219012
Fulltext PDF Free Access
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Abstract: For a Schwartz function on the plane and a nonzero define the Hilbert transform of in the direction to be p.v. Let be a Schwartz function with frequency support in the annulus , and . We prove that the maximal operator maps into weak , and into for . The estimate is sharp. The method of proof is based upon techniques related to the pointwise convergence of Fourier series. Indeed, our main theorem implies this result on Fourier series.
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 Bourgain, J., Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal., 1, 1991, 2, 147187. MR 1097257 (92g:42010)
 4.
 Carleson, Lennart, On convergence and growth of partial sums of Fourier series, Acta Math., 116, 1966, 135157. MR 0199631 (33:7774)
 5.
 Christ, Michael, Personal communication.
 6.
 Fefferman, Charles, Pointwise convergence of Fourier series, Ann. of Math. (2), 98, 1973, 551571. MR 0340926 (49:5676)
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 Grafakos, Loukas, Li, Xiaochun, Uniform bounds for the bilinear Hilbert transform, I, Ann. of Math., to appear.
 8.
 Grafakos, Loukas, Tao, Terence, Terwilleger, Erin, bounds for a maximal dyadic sum operator, Math. Z., 246, 2004, 12, 321337. MR 2031458 (2005a:42003)
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 Katz, Nets Hawk, Maximal operators over arbitrary sets of directions, Duke Math. J., 97, 1999, 1, 6779. MR 1681088 (2000a:42036)
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 Lacey, Michael, Carleson's Theorem: Proof, Complements, Variation, Pub. Mat., 48, 2004, no. 2, 251307. MR 2091007 (2005e:42020)
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Additional Information
Michael T. Lacey
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email:
lacey@math.gatech.edu
Xiaochun Li
Affiliation:
Department of Mathematics, University of California–Los Angeles, Los Angeles, California 900551555
Address at time of publication:
School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
Email:
xcli@math.ucla.edu
DOI:
http://dx.doi.org/10.1090/S0002994706038694
PII:
S 00029947(06)038694
Keywords:
Hilbert transform,
Fourier series,
maximal function,
pointwise convergence
Received by editor(s):
October 23, 2003
Received by editor(s) in revised form:
August 24, 2004
Published electronically:
March 24, 2006
Additional Notes:
The research of both authors was supported in part by NSF grants; the first author was also supported by the Guggenheim Foundation. Some of this research was completed during research stays by the first author at the Universite d’ParisSud, Orsay, and the Erwin Schrödinger Institute of Vienna Austria. The generosity of both is gratefully acknowledged.
Article copyright:
© Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
