# Maximal Ergodic Theorem On Weighted $L^p_w(X)$ spaces

## TL;DR

This paper establishes the boundedness of the maximal ergodic operator on weighted $L^p$ spaces with ergodic $A_p$ weights, extending classical ergodic theorems to weighted settings using transference techniques.

## Contribution

It introduces a new analysis of the maximal ergodic operator on weighted $L^p$ spaces with ergodic $A_p$ weights, employing transference methods.

## Key findings

- Boundedness of maximal ergodic operator on weighted spaces.
- Extension of ergodic theorems to weighted $L^p$ spaces.
- Use of transference method for weighted ergodic analysis.

## Abstract

In this paper, we study the maximal ergodic operator on $L^p_w(X, \mathcal{B}, \mu)$ spaces, $1 \leq p < \infty$, where $(X, \mathcal{B}, \mu)$ is a probability space equipped with an invertible measure preserving transformation $U$ and $w$ is an ergodic $A_p$ weight using transference method.