A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform

Lingxiao Zhang

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arXiv · 2605.04394

A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform

Lingxiao Zhang

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Abstract

The Zygmund vector-field maximal function conjecture is a long-standing open problem in harmonic analysis. It is notoriously difficult in part because the vector-field maximal function is non-translation-invariant and lies beyond the scope of finite-type theory. Its resolution would yield a Lebesgue differentiation theorem along vector fields, i.e., pointwise recovery of functions from averages along vector fields. We establish an $L^2$ boundedness criterion that substantially enlarges the class of admissible geometric degeneracies. More precisely, the power-type decay condition in Bourgain's theorem can be replaced by the much weaker logarithmic-polynomial decay $(\log\tfrac{1}{\tau})^{-p}$, $p>2$. This condition no longer yields the power-type integrability underlying Bourgain's argument, yet $L^2$ boundedness persists, permitting degeneracies far beyond any power-law regime. Our result also provides a unified geometric perspective on finite-type, intermediate-degenerate, and highly non-finite-type vector fields, with their behavior organized through the angular-variation quantity $w_x(t)$. This perspective places several classical boundedness criteria within a common hierarchy. Motivated by Lacey and Li's work on the Stein vector-field Hilbert transform conjecture, we further introduce a non-centered rectangular maximal operator adapted to the variable geometry of vector fields and prove its weak-type $(1,1)$ boundedness. Together, these results provide new tools and a unified geometric framework for the study of two fundamental conjectural problems in non-translation-invariant analysis.

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BibTeX RIS

Lingxiao Zhang. 2026-05-06. A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform. https://arxiv.org/abs/2605.04394

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