Estimates for periodic Zakharov–Shabat operators

We consider the periodic Zakharov–Shabat operators on the real line. The spectrum of this operator consists of intervals separated by gaps with the lengths |gn|⩾0, n∈Z. Let be the corresponding effective masses and let hn be heights of the corresponding slits in the quasi-momentum domain. We obtain a priori estimates of sequences g=(|gn|)n∈Z, , h=(hn)n∈Z in terms of weighted ℓp-norms at p⩾1. The proof is based on the analysis of the quasi-momentum as the conformal mapping.