A geometric proof of Bourgain's $L^2$ estimate of maximal operators along analytic vector fields

TL;DR

This paper provides a geometric proof of Bourgain's $L^2$ boundedness result for maximal operators along analytic vector fields, utilizing tools from Lacey and Li to offer a new perspective on the problem.

Contribution

It introduces a geometric proof of Bourgain's $L^2$ estimate, expanding the understanding of maximal operators along analytic vector fields.

Findings

Established $L^2$ boundedness of maximal operators along analytic vector fields

Developed a geometric proof approach based on Lacey and Li's tools

Enhanced theoretical understanding of maximal operators in harmonic analysis

Abstract

Bourgain proved that the maximal operator associated to an analytic vector field is bounded on $L^2$. In the present paper, we give a geometric proof of Bourgain's result by using the tools developed by Lacey and Li.