Application of Perron Trees to Geometric Maximal Operators

TL;DR

This paper investigates the boundedness of a geometric maximal operator related to rectangles with specific eccentricity and orientation, using generalized Perron trees to characterize $L^p$ bounds in $ ^2$.

Contribution

It introduces the use of generalized Perron trees to analyze the $L^p$ boundedness of the geometric maximal operator $M_{a,b}$ for rectangles with particular eccentricity and orientation.

Findings

Characterizes $L^p$ boundedness of $M_{a,b}$ for given $a,b>0$

Uses generalized Perron trees in the proof

Provides a geometric analysis of 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 $n \in \mathbb{N}^*$. The proof involves \textit{generalized Perron trees}, as constructed in \cite{KATHRYN JAN}.