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

Abstract

We introduce the notion of Perron capacity of a set of slopes Ω ⊂ R \Omega \subset \mathbb {R} . Precisely, we prove that if the Perron capacity of Ω \Omega is finite then the directional maximal operator associated M Ω M_\Omega is not bounded on L p ( R 2 ) L^p(\mathbb {R}^2) for any 1 > p > ∞ 1 > p > \infty . This allows us to prove that the set Ω c = { cos ⁡ n n : n ∈ N ∗ } \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.