# Proof of the square-function characterisation of L(log L)^{1/2}: part II

This is the 3rd post in a series that started with the post on the Chang-Wilson-Wolff inequality:

In today’s post we will finally complete the proof of the Tao-Wright lemma. Recall that in the 2nd post of the series we proved that the Tao-Wright lemma follows from its discrete version for Haar/dyadic-martingale-differences, which is as follows:

Lemma 2 – Square-function characterisation of $L(\log L)^{1/2}$ for martingale-differences:
For any function $f : [0,1] \to \mathbb{R}$ in $L(\log L)^{1/2}([0,1])$ there exists a collection $(F_j)_{j \in \mathbb{Z}}$ of non-negative functions such that:

1. for any $j \in \mathbb{N}$ and any $I \in \mathcal{D}_j$

$\displaystyle |\langle f, h_I \rangle|\lesssim \frac{1}{|I|^{1/2}} \int_{I} F_j \,dx;$

2. they satisfy the square-function estimate

$\displaystyle \Big\|\Big(\sum_{j \in \mathbb{N}} |F_j|^2\Big)^{1/2}\Big\|_{L^1} \lesssim \|f\|_{L(\log L)^{1/2}([0,1])}.$

Today we will prove this lemma.