A unified geometric perspective on Zygmund's conjecture for maximal functions associated with vector fields

Abstract

The Zygmund vector field maximal function conjecture is a long-standing open problem. This paper establishes a new boundedness criterion that significantly weakens the existing conditions in the literature. Specifically, the required decay condition is relaxed from the power-type decay of Bourgain for Zygmund's conjecture and the exponential-logarithmic decay of Lacey and Li for Stein's conjecture, to a logarithmic polynomial decay. Unlike the traditional framework that separates finite-type and non-finite-type operators, this paper offers a unified geometric view of both settings. The new criterion forms a natural continuation of a long-standing research line in harmonic analysis: it situates several pivotal conditions from earlier foundational works within a single developmental trajectory. Additionally, motivated by Lacey and Li's work, a non-centered rectangular maximal operator tailored to the underlying geometry is shown to satisfy weak $(1,1)$ boundedness. This operator serves as a novel tool for the subsequent study of the Zygmund and Stein conjectures.