论文标题
（UN）通过“ Perron容量”和应用程序的概念的定向最大运算符的界限
(Un)boundedness of directional maximal operators through a notion of "Perron capacity'' and an application
论文作者
论文摘要
我们介绍了一组斜率$ω\ subset \ mathbb {r} $的\ textIt {perron容量}的概念。确切地说，我们证明，如果$ω$的perron容量是有限的，那么对于任何$ 1 <p <\ iffty $，方向最大运算符$m_Ω$都不会在$ l^p（\ mathbb {r}^2）$上限制。这使我们能够证明$$ω_{\ boldsymbol {e}} = \ left \ {\ frac {\ frac {\ cos n} {n} {n} {n}：n \ in \ in \ mathbb {n}^*^* \ right \ right \} $ by在A. Stokolos提出的问题并不是一个有限的问题。
We introduce the notion of \textit{Perron capacity} of a set of slopes $Ω\subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $Ω$ is finite then the directional maximal operator associated $M_Ω$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$Ω_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.
