Littlewood–Paley theorem for arbitrary intervals: weighted estimates

Suppose $1<r<2$ and $b$ is a weight on $\mathbb R$ such that $b^{-\frac1{r-1}}$ satisfies the Muckenhoupt condition $A_{r'/2}$ ($r'$ is the exponent conjugate to $r$). If $f_j$ are functions whose Fourier transforms are supported on mutually disjoint intervals, then
$$ \Bigl\Vert\sum_j f_j\Bigr\Vert_{L^p(\mathbb R,b)}\le C\Bigl\Vert\Bigl(\sum_j|f_j|^2\Bigr)^{1/2}\Bigr\Vert_{L^p(\mathbb R,b)} $$
for $0<p\le r$. Bibl. – 9 titles.