A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform
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 L2 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 (log1/τ)−p, p>2. This condition no longer yields the power-type integrability underlying Bourgain's argument, yet L2 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 wx(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
@misc{zhang2026unified,
title = {{A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform}},
author = {Lingxiao Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.04394},
eprint = {2605.04394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.04394},
}
RIS
TY - UNPB
TI - A Unified Geometric Perspective on Zygmund's Maximal Function Conjecture and Related Questions for the Stein Hilbert Transform
AU - Lingxiao Zhang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2605.04394
UR - https://arxiv.org/abs/2605.04394
AB - 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 (log1/τ)^−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.
ER -