Littlewood-Paley theorem for arbitrary intervals: weighted estimates (Q1036960)

As stated by the author, the main result is as follows: ``Let \(1<r<2\) and let \(b\) be a weight on \({\mathbb R}\) such that \(b^{-1/(r-1)}\) 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 \[ \Big\| \sum_j f_j\Big\|_{L^p({\mathbb R},b)}\leq C \Big\| \Big(\sum_j |f_j|^2\Big)^{1/2}\Big\|_{L^p({\mathbb R},b)} \] for \(0<p\leq r\).''