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Media type:
E-Article
Title:
Maximal Theorems for the Directional Hilbert Transform on the Plane
Contributor:
Lacey, Michael T.;
Li, Xiaochun
imprint:
American Mathematical Society, 2006
Published in:Transactions of the American Mathematical Society
Language:
English
ISSN:
0002-9947
Origination:
Footnote:
Description:
<p>For a Schwartz function f on the plane and a non-zero <tex-math>$v \in \mathbb{R}^2$</tex-math> define the Hilbert transform of f in the direction v to be <tex-math>$H_{v}f(x) = p.v. \int_\mathbb{R} f(x - vy)_\frac{y} {dy}$</tex-math>. Let ζ be a Schwartz function with frequency support in the annulus <tex-math>$1 \leq \mid \xi \mid \leq 2$</tex-math>, and <tex-math>$\zeta f = \zeta * f$</tex-math>. We prove that the maximal operator <tex-math>$sup_{\mid v \mid = 1}\mid H_{v} \zeta f\mid$</tex-math> maps L<sup>2</sup> into weak L<sup>2</sup>, and L<sup>p</sup> into L<sup>p</sup> for p > 2. The L<sup>2</sup> 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.</p>