  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.
