Maximal theorems for the directional Hilbert transform on the plane
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- by Michael T. Lacey and Xiaochun Li PDF
- Trans. Amer. Math. Soc. 358 (2006), 4099-4117 Request permission
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 \begin{equation*} \operatorname H_vf(x)=\text {p.v.}\int _{\mathbb {R}} f(x-vy)\; \frac {dy}y. \end{equation*} Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le |\xi |\le 2$, and ${\boldsymbol \zeta }f=\zeta *f$. We prove that the maximal operator $\sup _{|v|=1}|\operatorname H_v{\boldsymbol \zeta } f|$ 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.References
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Additional Information
- Michael T. Lacey
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
- MR Author ID: 109040
- Email: lacey@math.gatech.edu
- 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
- Email: xcli@math.ucla.edu
- 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.
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 4099-4117
- MSC (2000): Primary 42B20, 42B25; Secondary 42B05
- DOI: https://doi.org/10.1090/S0002-9947-06-03869-4
- MathSciNet review: 2219012