On unboundedness of maximal operators for directional Hilbert transforms期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On unboundedness of maximal operators for directional Hilbert transforms

Authors:	G A Karagulyan

Institution:	Institute of Mathematics, Armenian National Academy of Sciences, Marshal Baghramian ave. 24b, Yerevan, 375019, Armenia

Abstract:	We show that for any infinite set of unit vectors in $\mathbb{R}^2$ the maximal operator defined by $\displaystyle H_Uf(x)=\sup_{u\in U}\bigg\vert{pv}\int_{-\infty }^\infty \frac{f(x-tu)}{t}dt\bigg\vert,\quad x\in \mathbb{R}^2,$ is not bounded in $L^2(\mathbb{R}^2)$ .

Keywords:	Hilbert transform maximal function

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