Application of Perron Trees to Geometric Maximal Operators

Abstract

We characterize the $L^p({\mathbb{R}}^2)$ boundeness of the geometric maximal operator $M_{a,b}$ associated to the basis ${\mathcal{B}}_{a,b}$ ($a,b>0$) which is composed of rectangles $R$ whose eccentricity and orientation is of the form $$\left(e_R,{\omega }_R\right)=\left(\frac{1}{n^a},\frac{\pi }{4n^b}\right)$$ for some $n\in {\mathbb{N}}^{\ast }$. The proof involves \textit{generalized Perron trees}, as constructed in \cite{KATHRYN JAN}.

arXiv admin note: text overlap with arXiv:2204.00253