Sharp threshold for uniform ℓ^1 boundedness

Determine the largest value of τ such that, for every η>0 and every F∈ℓ^2 with outer a_F and ||b_F||_{A_1}<τ−η, the ℓ^1 norm ||F||_{ℓ^1} is bounded by a constant depending only on η.

Background

The paper proves a positive quantitative Baxter estimate when ||b_F||{A_1}<1/\sqrt{3}−ε and constructs counterexamples showing that no uniform bound can hold once ||b_F||{A_1} can approach 1. Consequently, the unknown threshold τ lies in the interval 1/\sqrt{3}≤τ≤1.

The question asks for the exact endpoint separating a regime in which the A_1-smallness of b_F forces uniformly bounded ℓ1 potentials from a regime in which the paper’s counterexample mechanism prevents such control. The formulation assumes that a_F is outer on the exterior disk.

References

What is the largest value of $\tau$ such that for any $\eta>0$ and any $F\in \ell2()$ with $|b_F|{A_1}<\tau-\eta$ and $a_F$ outer on $*$, we have \begin{equation*} |F|{\ell1} \lesssim_{\eta} 1 \, ? \end{equation*}

— Quantitative possibilities for Baxter's theorem in nonlinear Fourier analysis  (2610.01711 - Alexis et al., 1 Oct 2026) in Question 3, labeled \texttt{quest\_theshold}, Section 1 (Introduction)