Hankel Schur multipliers and multipliers of $H^1$

Let $V^2$ be the space of matrices $\alpha=\{\alpha_{mk}\}_{m,k\geqslant0}$ such that $\sup_N\|P_N\alpha\|_{l^\infty\hat\otimes l^\infty}\leqslant\mathrm {const}$, where
\begin{gather*} (P_N\alpha)_{mk}=\begin{cases} \alpha_{mk},\quad m, k\leqslant N\\ 0,\quad\text{otherwise.} \end{cases} \end{gather*}
The papers concerns Hankel matrices (i. e. matrices $\Gamma_F=\{\hat F(m+k)_{m,k\geqslant0}\}$) in $V^2$. The following assertion is the main result of the paper.
Let $s_{j+1}/s_j\geqslant q>1$, $\{F_j\}_{j\geqslant1}$ be a sequence of polynomials whose Fourier coeeficients are supported on $[s_j, s_{j+1})$. If there exists a positive function $\omega$ on $\mathbb T$ such that $1/\omega\in L^1$ and $\sup_j\int_{\mathbb T}|F_j|^2\omega\,dm<\infty$ then $\Gamma_F\in V^2$.
As a corollary it is proved that under the above hypothesis $F$ is a multiplier of $H^1$, i. e.
$$ \varphi\in H^1\Rightarrow\varphi*F=\sum_{n\geqslant0}\hat\varphi(n)\hat F(n)z^n\in H^1. $$