Mockenhaupt的三项Hardy-Littlewood优函数猜想
Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture
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机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过解析延拓和Bessel积分表示，统一证明Mockenhaupt三项Hardy-Littlewood优函数猜想对所有k≥0成立，解决了k≥6的未决情形。
AI中文摘要：
对于整数$k \ge 0$，在$\mathbb{T} = \mathbb{R}/\mathbb{Z}$上定义$f_k(x) = 1 + e(x) + e((k+2)x)$和$g_k(x) = 1 + e(x) - e((k+2)x)$，其中$e(x) = e^{2\pi i x}$。Mockenhaupt猜想：当$2k < p < 2k+2$时，$\\|g_k\\|_{L^p(\mathbb{T})} > \\|f_k\\|_{L^p(\mathbb{T})}$。该猜想此前仅对$k \le 5$成立。我们给出一个对每个$k \ge 4$均有效的单一解析证明；特别地，这解决了所有先前未解决的$k \ge 6$情形，并确立了该猜想对每个$k \ge 0$成立。证明将范数比较归结为二维环面上的共振傅里叶系数，并在解析延拓后将这些系数表示为三重Bessel积分。Neumann乘积公式和Weber-Schafheitlin公式给出主导模态的定量正下界，而其余奇模态由一致尾部估计控制。随后证明主导模态对每个$k \ge 4$支配尾部。AI使用声明：本文的数学论证由自动研究系统Apex Math（Apex Intelligence构建的AI系统）生成。完整的AI使用声明见附录A。
英文摘要：
For an integer $k \ge 0$, let $f_k(x) = 1 + e(x) + e((k+2)x)$ and $g_k(x) = 1 + e(x) - e((k+2)x)$ on $\mathbb{T} = \mathbb{R}/\mathbb{Z}$, where $e(x) = e^{2πi x}$. Mockenhaupt conjectured that $\|g_k\|_{L^p(\mathbb{T})} > \|f_k\|_{L^p(\mathbb{T})}$ whenever $2k < p < 2k+2$. The conjecture was previously known for $k \le 5$. We give a single analytic proof valid for every $k \ge 4$; in particular, this settles all previously open cases $k \ge 6$ and establishes the conjecture for every $k \ge 0$. 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 $k \ge 4$. 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.
