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Maximal theorems for the directional Hilbert transform on the plane

Authors: Michael T. Lacey and Xiaochun Li
Journal: Trans. Amer. Math. Soc. 358 (2006), 4099-4117
MSC (2000): Primary 42B20, 42B25; Secondary 42B05
Published electronically: March 24, 2006
MathSciNet review: 2219012
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Abstract: For a Schwartz function $ f$ on the plane and a non-zero $ v\in\mathbb{R}^2$ define the Hilbert transform of $ f$ in the direction $ v$ to be

$\displaystyle \operatorname H_vf(x)=$p.v.$\displaystyle \int_{\mathbb{R}} f(x-vy)\; \frac{dy}y. $

Let $ \zeta$ be a Schwartz function with frequency support in the annulus $ 1\le\vert\xi\vert\le2$, and $ {\boldsymbol \zeta}f=\zeta*f$. We prove that the maximal operator $ \sup_{\vert v\vert=1}\vert\operatorname H_v{\boldsymbol \zeta} f\vert$ maps $ L^2$ into weak $ L^2$, and $ L^p$ into $ L^p$ for $ p>2$. The $ L^2$ 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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Additional Information

Michael T. Lacey
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Xiaochun Li
Affiliation: Department of Mathematics, University of California–Los Angeles, Los Angeles, California 90055-1555
Address at time of publication: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

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’Paris-Sud, 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.

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