Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function f on the plane and a non-zero v is an element of R-2 define the Hilbert transform of f in the direction v to be H-v f( x) = p. v. integral(R) f(x - vy) dy/y. Let zeta be a Schwartz function with frequency support in the annulus 1 <= |xi| <= 2, and zeta f = zeta * f. We prove that the maximal operator sup(|v|= 1) |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. Indeed, our main theorem implies this result on Fourier series.