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arXiv · 2204.00253

Kakeya-type sets for Geometric Maximal Operators

Anthony Gauvan
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Abstract

Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $\lambda$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($\lambda$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $\Omega$ is an arbitrary set of angles in [0, $\pi$ 4), we prove that any rarefied basis B of the directional basis R $\Omega$ yields an operator MB that has the same L p-behavior than the directional maximal operator M $\Omega$ for 1 < p < $\infty$.

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Anthony Gauvan. 2022-04-01. Kakeya-type sets for Geometric Maximal Operators. https://arxiv.org/abs/2204.00253

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