Mockenhaupt's Three-Term Hardy-Littlewood Majorant Conjecture

Abstract

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 $e(x)=e^{2\pi ix}$. Mockenhaupt conjectured that $\Vert g_k{\Vert }_{L^p(\mathbb{T})}>\Vert f_k{\Vert }_{L^p(\mathbb{T})}$ 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.

v2: substantially revised and shortened; 23 pages, was 35. Title changed. The binomial-moment reduction of former Section 2 is replaced by a direct argument, the torus normalisation changed to the e(.) convention, and the sections on endpoint exponents, small cases and the symbol index removed. References and acknowledgment updated. Ancillary scripts rewritten for this version