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Probabilistic Construction of Kakeya-Type Sets in $\mathbb{R}^2$ associated to separated sets of directions
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abstract
We provide a condition on a set of directions $\Omega \subset \mathbb{S}^1$ ensuring that the associated directional maximal operator $M_\Omega$ is unbounded on $L^p(\mathbb{R}^2)$ for every $1 \leq p < \infty$. 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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On differentiation of integrals in Lebesgue spaces
For every p0, there are rectangle-based Busemann-Feller bases on T^ω whose L^p differentiation range is exactly [p0,∞] or (p0,∞], and these ranges exhaust the six possible forms for complete metric measure spaces.
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