Maximal Theorems for the Directional Hilbert Transform on the Plane

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.