Let ${��_q(n)}_{n \in \mathbb{N}}$ be the lengths of spectral gaps in a continuous spectrum of the Hill-Schr��dinger operators S(q)u=-u''+q(x)u,\quad x\in \mathbb{R}, with 1-periodic real-valued potentials $q \in L^{2}(\mathbb{T})$. Let weight function $��:\;[1,\infty)\to (0,\infty)$. We prove that under the condition \exists s\in [0,\infty):\quad k^{s}\ll��(k)\ll k^{s+1},\; k\in \mathbb{N}, the map $��:\, q \mapsto \{��_{q}(n)}_{n \in \mathbb{N}}$ satisfies the equalities: \verb"i")\quad ��(H^��) = h_{+}^��, \verb"ii")\quad ��^{-1}(h_{+}^��) = H^��, where the real function space H^�� & ={f=\sum_{k\in \mathbb{Z}}\hat{f}\,(k)e^{i k2��x}\in L^{2}(\mathbb{T})| \sum_{k\in \mathbb{N}} ��^{2}(k)|\hat{f}(k)|^{2}1,c>1:\qquad c^{-1}\leq \frac{��(��t)}{��(t)}\leq c\quad\forall t\geq 1,\;��\in [1,a] then the function class $H^��$ is a real H��rmander space $H_{2}^��(\mathbb{T},\mathbb{R})$ with the weight $��(\sqrt{1+��^{2}})$.

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