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

1. The Chang-Wilson-Wolff inequality using a lemma of Tao-Wright

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

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 for martingale-differences:

For any function in there exists a collection of non-negative functions such that:

- for any and any

- they satisfy the square-function estimate

Today we will prove this lemma.