Littlewood-Paley type inequality on ℝ

Abstract

Let {Ik}k∈ℕ be a sequence of well-distributed mutually disjoint intervals of ℝ\{0}. For f ∈ Lp(ℝ), 1 ≤ p ≤ 2, define SIkf by (SIkf)∧ = χIkikf̂ We prove that there exists a positive constant C such that

(Σk∈ℕ;|SIkf|p′)1/p′

p,p′ ≤ C

f

p for all f ∈ Lp(ℝ), 1 < p < 2, where 1/p + 1/p′ = 1 and

·

p,p′ is the norm of the Lorentz space Lp,p′ (ℝ). An application of our result to Fourier multipliers is given.