Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture

Abstract

For an integer k ≥ 0, let f_k(x) = 1 + e(x) + e((k+2)x) and g_k(x) = 1 + e(x) − e((k+2)x) on 𝕋 = ℝ/ℤ, where e(x) = e^{2πi x}. Mockenhaupt conjectured that |g_k|_L^p(𝕋) > |f_k|_L^p(𝕋) whenever 2k < p < 2k+2. The conjecture was previously known for k ≤ 5. We give a single analytic proof valid for every k ≥ 4; in particular, this settles all previously open cases k ≥ 6 and establishes the conjecture for every k ≥ 0. The proof reduces the norm comparison to resonant Fourier coefficients on the two-torus and represents these coefficients, after analytic continuation, by triple-Bessel integrals. Neumann's product formula and the Weber-Schafheitlin formula yield a quantitative positive lower bound for the leading mode, while the remaining odd modes are controlled by a uniform tail estimate. The leading mode is then shown to dominate the tail for every k ≥ 4. AI Usage. The mathematical argument of this paper was produced by the auto-research system Apex Math, an AI system built by Apex Intelligence. See Appendix A for the complete AI usage statement.

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BibTeX RIS

BibTeX

@misc{pan2026mockenhaupts,
  title = {{Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture}},
  author = {Guancheng Pan and Chengsong You and Hengyu Wang and Junwei Zhou and Wenjun Zhang and Yongchao Chen},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09740},
  eprint = {2609.09740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09740},
}

RIS

TY  - UNPB
TI  - Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture
AU  - Guancheng Pan
AU  - Chengsong You
AU  - Hengyu Wang
AU  - Junwei Zhou
AU  - Wenjun Zhang
AU  - Yongchao Chen
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.09740
UR  - https://arxiv.org/abs/2609.09740
AB  - For an integer k ≥ 0, let f_k(x) = 1 + e(x) + e((k+2)x) and g_k(x) = 1 + e(x) − e((k+2)x) on 𝕋 = ℝ/ℤ, where e(x) = e^2πi x. Mockenhaupt conjectured that |g_k|_L^p(𝕋) > |f_k|_L^p(𝕋) whenever 2k < p < 2k+2. The conjecture was previously known for k ≤ 5. We give a single analytic proof valid for every k ≥ 4; in particular, this settles all previously open cases k ≥ 6 and establishes the conjecture for every k ≥ 0. The proof reduces the norm comparison to resonant Fourier coefficients on the two-torus and represents these coefficients, after analytic continuation, by triple-Bessel integrals. Neumann's product formula and the Weber-Schafheitlin formula yield a quantitative positive lower bound for the leading mode, while the remaining odd modes are controlled by a uniform tail estimate. The leading mode is then shown to dominate the tail for every k ≥ 4. AI Usage. The mathematical argument of this paper was produced by the auto-research system Apex Math, an AI system built by Apex Intelligence. See Appendix A for the complete AI usage statement.
ER  -

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