A note on Littlewood-Paley decompositions with arbitrary intervals (Q1084606)

For every interval I in \({\mathbb{R}}\) define an operator \(S_ I\) by setting \((S_ If)^{\wedge}=\chi_ I\hat f\), where \(\hat f\) denotes the Fourier transform of f. Let \((I_ k)_ 1^{\infty}\) be a sequence of disjoint intervals and set \(\Delta f(x)=(\sum^{\infty}_{1}| S_{I_ k}f(x)|^ 2)^{1/2}.\) \textit{J. L. Rubio de Francia} has proved that \(\Delta\) is a bounded operator on \(L^ p({\mathbb{R}})\) for \(2\leq p<\infty\) [Report No.18, Institut Mittag-Leffler (1983)]. This paper contains an alternative proof of the basic inequality in the proof of the above theorem.