On unboundedness of maximal operators for directional Hilbert transforms
arXiv:math/0602524
Abstract
We show that for any infinite set of unit vectors in $\ZR^2$ the maximal operator defined by $$ H_Uf(x)=\sup_{u\in U}\bigg|\pv\int_{-\infty}^\infty \frac{f(x-tu)}{t}dt\bigg|,\quad x\in \ZR^2, $$ is not bounded in $L^2(\ZR^2)$.
Published in: Proceedings of the American Mathematical Society 135 (2007), no. 10, 3133-3141