Maximal theorems for the directional Hilbert transform on the plane

For a Schwartz function f on the plane and a non-zero v ∈ ℝ ² define the Hubert transform of f in the direction v to be H _v f(x) = p.v. ∫ _ℝ f(x-vy) dy/y. Let ζ be a Schwartz function with frequency support in the annulus 1 ≤ |ξ| ≤ 2, and ζf = ζ * f. We prove that the maximal operator sup _|v|=1 | H _vζf| 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.