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Maximal theorems for the directional Hilbert transform on the plane

Authors:	Michael T Lacey Xiaochun Li

Institution:	School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 ; Department of Mathematics, University of California--Los Angeles, Los Angeles, California 90055-1555

Abstract:	For a Schwartz function on the plane and a non-zero $v\in\mathbb{R}^2$ define the Hilbert transform of in the direction 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 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.

Keywords:	Hilbert transform Fourier series maximal function pointwise convergence

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