q-Littlewood-Paley g-function

Littlewood-Paley g-Functions and Multipliers for the Laguerre Hypergroup

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.

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

On Bilinear Littlewood-paley Square Functions

On the real line, let the Fourier transform of kn be k̂n(ξ) = k̂(ξ−n) where k̂(ξ) is a smooth compactly supported function. Consider the bilinear operators Sn(f, g)(x) = ∫ f(x + y)g(x − y)kn(y) dy. If 2 ≤ p, q ≤ ∞, with 1/p + 1/q = 1/2, I prove that ∞ ∑ n=−∞ ‖Sn(f, g)‖2 ≤ C‖f‖p‖g‖q . The constant C depends only upon k.