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On unboundedness of maximal operators for directional Hilbert transforms

Research paper by G. A. Karagulyan

Indexed on: 07 Sep '09Published on: 07 Sep '09Published in: Mathematics - Classical Analysis and ODEs



Abstract

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 $L^2(\ZR^2)$.