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.
Categories: math.SP math.CV