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

Abstract

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 $${\Omega }_{\mathbf{e}}=\left\{\frac{\cos n}{n}:n\in {\mathbb{N}}^{\ast }\right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.