On a Littlewood-Paley type inequality

Littlewood–Paley Inequality: A Survey

Let Sωf = ∫ ω f̂(ξ)e ixξ dξ be the Fourier projection operator to an interval ω in the real line. Rubio de Francia’s Littlewood Paley inequality [31] states that for any collection of disjoint intervals Ω, we have ∥∥ [∑ ω∈Ω |Sωf | 1/2∥∥ p . ‖f‖p, 2 ≤ p < ∞. We survey developments related to this inequality, including the higher dimensional case, and consequences for multipliers.

A Semi-discrete Littlewood-paley Inequality

Rubio de Francia’s Littlewood-Paley inequality for operator-valued functions

We prove Rubio de Francia’s Littlewood-Paley inequality for arbitrary disjoint intervals in the noncommutative setting, i.e. for functions with values in noncommutative L-spaces. As applications, we get sufficient conditions in terms of q-variation for the boundedness of Schur multipliers on Schatten classes.

Issues related to Rubio de Francia’s Littlewood–Paley inequality

Let Sω f = ∫ ω f̂(ξ)e dξ be the Fourier projection operator to an interval ω in the real line. Rubio de Francia’s Littlewood–Paley inequality (Rubio de Francia, 1985) states that for any collection of disjoint intervals Ω, we have ∥∥∥∥ [∑ ω∈Ω |Sω f | 1/2∥∥∥∥ p ‖f‖p, 2 ≤ p < ∞. We survey developments related to this inequality, including the higher dimensional case, and consequences for multiplie...

Littlewood–Paley Inequailty: A Survey

Let Sωf = ∫ ω f̂(ξ)e ixξ dξ be the Fourier projection operator to an interval ω in the real line. Rubio de Francia’s Littlewood Paley inequality [28] states that for any collection of disjoint intervals Ω, we have ∥∥ [∑ ω∈Ω |Sωf | 1/2∥∥ p . ‖f‖p, 2 ≤ p < ∞. We survey developments related to this inequality, including the higher dimensional case, and consequences for multipliers.