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 $|g_n|\ge 0, n\in \Z$. Let $\m_n^\pm$ be the corresponding effective masses and let $h_n$ be heights of the corresponding slits in the quasimomentum domain. We obtain a priori estimates of sequences $g=(|g_n|)_{n\in \Z},\m^\pm=(\m_n^\pm)_{n\in \Z}, h=(h_n)_{n\in \Z}$ in terms of weighted $\ell^p-$norms at $p\ge 1$. The proof is based on the analysis of the quasimomentum as the conformal mapping. 
Evgeny Korotyaev and Pavel Kargaev