On maximal functions associated with planar vector fields

TL;DR

This paper refines Bourgain's method to establish weaker conditions for the boundedness of maximal functions linked to planar vector fields, strengthening prior results and comparing different boundedness criteria.

Contribution

It introduces a weaker condition for boundedness, providing an elementary proof and extending the understanding of related operators in various settings.

Findings

Identifies a weaker boundedness condition for maximal functions

Strengthens a result implicit in Lacey and Li's work

Provides an elementary proof following Bourgain's original approach

Abstract

By refining Bourgain's argument for maximal functions associated with planar vector fields, we identify a condition ensuring boundedness that is weaker than previously known. As a consequence, this strengthens a result implicit in the work of Lacey and Li. The proof is elementary and follows Bourgain's original method. In addition, we compare boundedness criteria in both finite-type and non-finite-type settings for related operators.