On unboundedness of maximal operators for directional Hilbert transforms

Karagulyan, G.

doi:10.1090/S0002-9939-07-08731-X

On unboundedness of maximal operators for directional Hilbert transforms
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Proc. Amer. Math. Soc. 135 (2007), 3133-3141 Request permission

Abstract:

We show that for any infinite set of unit vectors $U$ in $\mathbb {R}^2$ the maximal operator defined by \begin{equation*} H_Uf(x)=\sup _{u\in U}\bigg |\operatorname {pv}\int _{-\infty }^\infty \frac {f(x-tu)}{t}dt\bigg |,\quad x\in \mathbb {R}^2, \end{equation*} is not bounded in $L^2(\mathbb {R}^2)$.

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