The qNCC-I quantitative Nonlinear Carleson Conjecture

Let \boldsymbol{\textstyle\gamma}=\\{\gamma_n\}_{n\in\mathbb{Z}_+} be the sequence of Verblunsky coefficients associated with the measure $\sigma$, and let $M(\xi,\sigma)$ denote the maximal function introduced above. qNCC-I. There is $\epsilon>0$ such that

$$\|\{\gamma_n\}\|_{\ell^2(\mathbb{Z}_+)}\le \epsilon$$

implies

$$\int_{\mathbb{T}}M^2(\xi,\sigma)\,d\sigma\lesssim \|\{\gamma_n\}\|^2_{\ell^2(\mathbb{Z}_+)}.$$

This is a quantitative strengthening of the Nonlinear Carleson Conjecture, proposing an $L^2$ bound for the nonlinear maximal function under a smallness assumption on the Verblunsky coefficients. Its resolution is not supplied in the source, so it remains open.