Maximal Theorems for the Directional Hilbert Transform on the Plane

Details Maximal Theorems for the Directional Hilbert Transform on the Plane Maximal Theorems for the Directional Hilbert Transform on the Plane

2003-10-21

arxiv.org/abs/math/0310346v2

Mathematics

Classical Analysis and ODEs

Substantially revised with 23 pages, 8 figures, and 14 references

Scientific paper

For a Schwartz function $f$ on the plane and a non-zero $v\in\ZR^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$ H_vf(x)=\text{p.v.}\int_\ZR f(x-vy) \frac{dy}y $$ Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le| \xi|\le2$. We prove that the maximal operator $$ \sup_{\abs v=1}\abs{H_v{\zeta}* f} $$ 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, especially the recent proof given by Lacey and Thiele.

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