SU(1,1) nonlinear Fourier bound by the nonlinear coefficient

Determine whether every sequence F and its SU(1,1)-valued nonlinear Fourier transform (a_F,b_F) satisfy the estimate ||F||_{ℓ^1} \lesssim ||b_F||_{A_1}.

Background

The paper studies quantitative analogues of Baxter’s theorem for nonlinear Fourier transforms. In the SU(1,1) setting, the classical qualitative theorem states that F belongs to ℓ1 if and only if the nonlinear Fourier coefficients a_F and b_F belong to the Wiener algebra A_1.

The authors ask whether the ℓ1 norm of the potential can be controlled solely by the A_1 norm of b_F. They note that their counterexample showing that ||F||{ℓ1} cannot be controlled by ||a_F||{A_1} applies to SU(1,1), but does not settle the proposed estimate involving b_F.

References

For a sequence $F$ and its $SU(1,1)$-NLFT $(a_F,b_F)$, do we have \begin{equation}\label{eq:counterexampe su11 F and b} |F|{\ell1} \lesssim |b_F|{A_1} \, ? \end{equation}

eq:counterexampe su11 F and b:

∥F∥ℓ1≲∥bF∥A1 ?\|F\|_{\ell^1} \lesssim \|b_F\|_{A_1} \, ?

— Quantitative possibilities for Baxter's theorem in nonlinear Fourier analysis  (2610.01711 - Alexis et al., 1 Oct 2026) in Question in Section 1, immediately following Theorem 2 (Introduction)

For $\eta>0$, is it true that for any $F\in \ell2()$ with $|F|{\ell\infty}<1-\eta$ and $a_F$ outer on $*$, we have \begin{equation*} |F|{\ell1} \lesssim_{\eta} |b_F|_{A_1} \, ? \end{equation*}

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