Littlewood–Paley type inequality on ℝ

Let { I k } k ∈ℕ be a sequence of well–distributed mutually disjoint intervals of ℝ\{0}. For f ∈ L p (ℝ), 1 ≤ p ≤ 2, define S   I   kf by ( S   I   kf )ˆ = χ   I   k$\hat f $ . We prove that there exists a positive constant C such that\documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document} $$ \big\Vert \big( \sum _{k \in \mathbb {N}} \big\vert S _{I_{k}}f \big\vert ^{p \prime} \big) ^{1/p \prime} \big\Vert _{p,p \prime} \le C \vert f \vert _{p} $$ \end{document}for all f ∈ L p (ℝ), 1 < p < 2, where 1/ p + 1/ p ′ = 1 and ‖ · ‖ p,p ′ is the norm of the Lorentz space L p,p ′ (ℝ). An application of our result to Fourier multipliers is given.