Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture

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 $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$, where $e(x)=e^{2\pi ix}$. Mockenhaupt conjectured that $\mathrm{\parallel }{g}_k{\mathrm{\parallel }}_{L^p(\mathbb{T})}>\mathrm{\parallel }{f}_k{\mathrm{\parallel }}_{L^p(\mathbb{T})}$ 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.