(Un)boundedness of directional maximal operators through a notion of “Perron capacity” and an application

D’Aniello, Emma; Gauvan, Anthony; Moonens, Laurent

doi:10.1090/proc/16291

(Un)boundedness of directional maximal operators through a notion of “Perron capacity” and an application
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by Emma D’Aniello, Anthony Gauvan and Laurent Moonens

Proc. Amer. Math. Soc. 151 (2023), 2517-2526

DOI: https://doi.org/10.1090/proc/16291

Published electronically: March 14, 2023

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Abstract:

We introduce the notion of Perron capacity of a set of slopes $\Omega \subset \mathbb {R}$. Precisely, we prove that if the Perron capacity of $\Omega$ is finite then the directional maximal operator associated $M_\Omega$ is not bounded on $L^p(\mathbb {R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set \begin{equation*} {\Omega _c} =\left \{ \frac {\cos n}{n}: n\in \mathbb {N}^* \right \} \end{equation*} is not finitely lacunary which answers a question raised by A. Stokolos in private communications to the authors.

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