Two quantitative versions of the Nonlinear Carleson Conjecture

TL;DR

The paper investigates two quantitative formulations of the Nonlinear Carleson Conjecture (NCC) in the setting of orthogonal polynomials on the unit circle (OPUC) and $SU(1,1)$ matrix products. It defines $\text{qNCC-I}$ and $\text{qNCC-II}$ via $L^2$ control of the maximal nonlinear Fourier data $M(\xi,\sigma)$ and the logarithmic bound on $O(\xi)$, proves $\text{qNCC-I} \Rightarrow \text{qNCC-II}$, and grounds these conjectures in Menshov–Rademacher theory and variational matrix-product methods. The work derives significant consequences, including a.e. convergence of $\Pi_n(\xi)$ and $\phi_n^*(\xi,\sigma)$ to Szegő objects, a BD-type characterization of zeros of $\phi_n$, and connections to Carleson–Hunt maximal bounds via weakened NCC versions. It also develops weak NCC variants (wqNCC) and shows how small $\ell^2$-norms of Schur parameters yield Carleson-type estimates, thereby linking nonlinear Fourier analysis with classical harmonic analysis and variational techniques. Overall, the two quantified NCCs provide a framework that implies NCC and bridges nonlinear Fourier transforms with maximal-function theory and Calderón–Stein-type results.

Abstract

This note compares two quantitative versions of the Nonlinear Carleson Conjecture (NCC). We provide motivations for our conjectures and show that they both imply the NCC. We discuss the connection to Carleson-Hunt maximal functions and give an SU(1,1) version of Calderon's theorem.

Theorems & Definitions (57)

• Theorem 2.1
• proof
• Theorem 2.2: Menshov-Rademacher
• Theorem 3.1
• proof
• Theorem 3.2: Bessonov-Denisov, BD
• Corollary 3.3
• Theorem 4.1
• proof
• Theorem 4.2