Maximal theorems for the directional Hilbert transform on the plane

Abstract:

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 \begin{equation*} \operatorname H_vf(x)=\text {p.v.}\int _{\mathbb {R}} f(x-vy)\; \frac {dy}y. \end{equation*} Let $\zeta$ be a Schwartz function with frequency support in the annulus $1\le |\xi |\le 2$, and ${\boldsymbol \zeta }f=\zeta *f$. We prove that the maximal operator $\sup _{|v|=1}|\operatorname H_v{\boldsymbol \zeta } f|$ 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.