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, . 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 , , in terms of weighted ℓp-norms at p 1. The proof is based on the analysis of the quasi-momentum as the conformal mapping.