(Un)boundedness of directional maximal operators through a notion of 'Perron capacity' and an application
| Autor: | D'Aniello, Emma, Gauvan, Anthony, Moonens, Laurent |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | We introduce the notion of \textit{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 $$\Omega_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos. |
| Databáze: | arXiv |
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