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 ?p-norms at p?1. The proof is based on the analysis of the quasi-momentum as the conformal mapping.