Perron's capacity of random sets

Anthony Gauvan

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arXiv · 2312.11964

Perron's capacity of random sets

Anthony Gauvan

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Abstract

Given a sequence of random variables $\left\{ X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions $$\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}$$ and $$\Omega_{\text{rand},\text{lac}} := \left\{ \frac{ \pi X_k}{2^k} : k\geq 1 \right\}.$$ We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$.

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BibTeX RIS

Anthony Gauvan. 2023-12-19. Perron's capacity of random sets. https://arxiv.org/abs/2312.11964

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