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We provide a condition on a set of directions S¹ ensuring that the associated directional maximal operator M_ is unbounded on Lᵖ (R²) for every 1 p <. The techniques of proof extend ideas of Bateman and Katz involving probabilistic construction of Kakeya-type sets involving sticky maps and Bernoulli percolation.
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Hagelstein et al. (Mon,) studied this question.
synapsesocial.com/papers/68e68593b6db64358760e037 — DOI: https://doi.org/10.48550/arxiv.2405.17674
Paul Hagelstein
Blanca Radillo-Murguia
Alexander M. Stokolos
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