Maximal theorems for the directional Hilbert transform on the plane

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

$\operatorname H_vf(x)=$p.v.$\int_{\mathbb{R}} f(x-vy)\; \frac{dy}y.$

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 $L^2$ 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. Indeed, our main theorem implies this result on Fourier series.