On the Littlewood–Paley theorem for arbitrary intervals

We extend the results of Rubio de Francia [1] and Bourgain [2] by showing that for arbitrary mutually nonintersecting intervals $\Delta_k\subset\mathbb Z_+$, arbitrary $p\in(0,2]$, and arbitrary trigonometric polynomials $f_k$ with $\mathrm{supp}\,\widehat f_k\subset\Delta_k$, we have
$$ \biggl\|\sum_k f_k\biggr\|_{H^p(\mathbb T)}\le a_p\biggl\|\biggl(\sum_k|f_k|^2\biggr)^{1/2}\biggr\|_{L^p(\mathbb T)}. $$
The method is a development of that by Rubio de Francia.