Maximal Theorems for the Directional Hilbert Transform on the Plane

For a Schwartz function $f$ on the plane and a non-zero $v\in {\text{ZR}}^2$ define the Hilbert transform of $f$ in the direction $v$ to be $$H_{v}f(x)=\text{p.v.}{\int }_{\text{ZR}}f(x-vy)\frac{dy}{y}$$ Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le \mathrm{\mid }\xi \mathrm{\mid }\le 2$. We prove that the maximal operator $$\underset{\text{abs}v=1}{\sup }\text{abs}{H}_v\zeta \ast f$$ 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.