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:

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 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.

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