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On directional maximal operators associated with generalized lacunary sets

G. A. Karagulyan
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Abstract

Let $Ω$ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_Ωf(x)=\sup_{δ>0, ω\in Ω} \frac{1}{2δ}\int_{-δ}^δ|f(x+tω)|dt. \emd for the generalized lacunary sets $Ω$ associated with an integer $μ>0$. It is proved the following sharp inequality: $$ \|M_Ωf(x)\|_2\lesssim \sqrtμ \|f\|_2. $$

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G. A. Karagulyan. 2005-11-19. On directional maximal operators associated with generalized lacunary sets. https://arxiv.org/abs/math/0511480

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