On unboundedness of maximal operators for directional Hilbert transforms

Details On unboundedness of maximal operators for directional Hilbert transforms On unboundedness of maximal operators for directional Hilbert transforms

2006-02-23

arxiv.org/abs/math/0602524v2

Mathematics

Classical Analysis and ODEs

Published in: Proceedings of the American Mathematical Society 135 (2007), no. 10, 3133-3141

Scientific paper

We show that for any infinite set of unit vectors $U$ 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 $L^2(\ZR^2)$.

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