Mockenhaupt’s Three-Term Hardy-Littlewood Majorant Conjecture

Authors: Guancheng Pan, Chengsong You, Hengyu Wang, Junwei Zhou, Yongchao Chen

Preprint: arXiv:2609.09740

Abstract

This paper settles Mockenhaupt’s three-term Hardy-Littlewood majorant conjecture on the torus: the norm-comparison inequality holds whenever (2k < p < 2k+2). The statement was previously known only for (k \leq 5); the authors give a single analytic proof valid for every (k \geq 4), settling all previously open cases (k \geq 6) and confirming the conjecture for all (k \geq 0). The argument reduces the comparison to resonant Fourier coefficients on the two-torus, represents them via triple-Bessel integrals after analytic continuation, and bounds the leading mode using Neumann’s product formula together with the Weber-Schafheitlin formula against a controlled tail.

Notably, the proof was produced by Apex Math, an auto-research system built by Apex Intelligence.

See the official arXiv record for the full paper and citation details.