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$\ell^1$ mapping properties, smoothness and decay for $SU(2)$-valued nonlinear Fourier transform

Gevorg Mnatsakanyan

Abstract

We prove an analog of Baxter's theorem for $SU(2)$-valued nonlinear Fourier transform (NLFT). That is, we prove that under certain natural conditions on the NLFT data, the potential is in $\ell^1$ if and only if the linear Fourier coefficients of the NLFT data are in $\ell^1$. Furthermore, we prove some smoothness-decay estimates for the NLFT motivated by similar estimates for the linear Fourier transform.

$\ell^1$ mapping properties, smoothness and decay for $SU(2)$-valued nonlinear Fourier transform

Abstract

We prove an analog of Baxter's theorem for -valued nonlinear Fourier transform (NLFT). That is, we prove that under certain natural conditions on the NLFT data, the potential is in if and only if the linear Fourier coefficients of the NLFT data are in . Furthermore, we prove some smoothness-decay estimates for the NLFT motivated by similar estimates for the linear Fourier transform.
Paper Structure (9 sections, 9 theorems, 120 equations)

This paper contains 9 sections, 9 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on $SU(2)$-nonlinear Fourier transform
  3. Riemann-Hilbert factorization
  4. Proof of Theorem \ref{['main thm']}
  5. The easy part
  6. Some Notation and Preliminaries
  7. The difficult part
  8. Proof of Proposition \ref{['quantitative root 2']}
  9. Proof of Theorem \ref{['thm decay']}

Key Result

Theorem 1

Assume $(a,b)$ is the NLFT image of the sequence $F=(F_j)_{j\in \mathbb Z} \in \ell^2 (\mathbb Z)$ and $w$ is a strong Beurling weight. If $F\in \ell^1_w$, then $a^*,b \in A_w$. Conversely, if $a^*$ is outer on $\mathbb D$ and $b/a^* \in A_w$, then $F\in \ell_w^1$.

Theorems & Definitions (15)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Lemma 1: Wiener's lemma for $A_w$
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 5 more