# 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.

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

This is a follow-up on the post on the Chang-Wilson-Wolff inequality and how it can be proven using a lemma of Tao-Wright. The latter consists of a square-function characterisation of the Orlicz space $L(\log L)^{1/2}$ analogous in spirit to the better known one for the Hardy spaces.
In this post we will commence the proof of the Tao-Wright lemma, as promised. We will start by showing how the lemma, which is stated for smooth frequency projections, can be deduced from its discrete variant stated in terms of Haar coefficients (or equivalently, martingale differences with respect to the standard dyadic filtration). This is a minor part of the overall argument but it is slightly tricky so I thought I would spell it out.

Recall that the Tao-Wright lemma is as follows. We write $\widetilde{\Delta}_j f$ for the smooth frequency projection defined by $\widehat{\widetilde{\Delta}_j f}(\xi) = \widehat{\psi}(2^{-j}\xi) \widehat{f}(\xi)$, where $\widehat{\psi}$ is a smooth function compactly supported in $1/2 \leq |\xi| \leq 4$ and identically equal to 1 on $1 \leq |\xi| \leq 2$.

Lemma 1 – Square-function characterisation of $L(\log L)^{1/2}$ [Tao-Wright, 2001]:
Let for any $j \in \mathbb{Z}$

$\displaystyle \phi_j(x) := \frac{2^j}{(1 + 2^j |x|)^{3/2}}$

(notice $\phi_j$ is concentrated in $[-2^{-j},2^{-j}]$ and $\|\phi_j\|_{L^1} \sim 1$).
If the function ${f}$ is in $L(\log L)^{1/2}([-R,R])$ and such that $\int f(x) \,dx = 0$, then there exists a collection $(F_j)_{j \in \mathbb{Z}}$ of non-negative functions such that:

1. pointwise for any $j \in \mathbb{Z}$

$\displaystyle \big|\widetilde{\Delta}_j f\big| \lesssim F_j \ast \phi_j ;$

2. they satisfy the square-function estimate

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

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

Today I would like to introduce an important inequality from the theory of martingales that will be the subject of a few more posts. This inequality will further provide the opportunity to introduce a very interesting and powerful result of Tao and Wright – a sort of square-function characterisation for the Orlicz space $L(\log L)^{1/2}$.

## 1. The Chang-Wilson-Wolff inequality

Consider the collection $\mathcal{D}$ of standard dyadic intervals that are contained in $[0,1]$. We let $\mathcal{D}_j$ for each $j \in \mathbb{N}$ denote the subcollection of intervals $I \in \mathcal{D}$ such that $|I|= 2^{-j}$. Notice that these subcollections generate a filtration of $\mathcal{D}$, that is $(\sigma(\mathcal{D}_j))_{j \in \mathbb{N}}$, where $\sigma(\mathcal{D}_j)$ denotes the sigma-algebra generated by the collection $\mathcal{D}_j$. We can associate to this filtration the conditional expectation operators

$\displaystyle \mathbf{E}_j f := \mathbf{E}[f \,|\, \sigma(\mathcal{D}_j)],$

and therefore define the martingale differences

$\displaystyle \mathbf{D}_j f:= \mathbf{E}_{j+1} f - \mathbf{E}_{j}f.$

With this notation, we have the formal telescopic identity

$\displaystyle f = \mathbf{E}_0 f + \sum_{j \in \mathbb{N}} \mathbf{D}_j f.$

Demystification: the expectation $\mathbf{E}_j f(x)$ is simply $\frac{1}{|I|} \int_I f(y) \,dy$, where $I$ is the unique dyadic interval in $\mathcal{D}_j$ such that $x \in I$.

Letting $f_j := \mathbf{E}_j f$ for brevity, the sequence of functions $(f_j)_{j \in \mathbb{N}}$ is called a martingale (hence the name “martingale differences” above) because it satisfies the martingale property that the conditional expectation of “future values” at the present time is the present value, that is

$\displaystyle \mathbf{E}_{j} f_{j+1} = f_j.$

In the following we will only be interested in functions with zero average, that is functions such that $\mathbf{E}_0 f = 0$. Given such a function $f : [0,1] \to \mathbb{R}$ then, we can define its martingale square function $S_{\mathcal{D}}f$ to be

$\displaystyle S_{\mathcal{D}} f := \Big(\sum_{j \in \mathbb{N}} |\mathbf{D}_j f|^2 \Big)^{1/2}.$

With these definitions in place we can state the Chang-Wilson-Wolff inequality as follows.

C-W-W inequality: Let ${f : [0,1] \to \mathbb{R}}$ be such that $\mathbf{E}_0 f = 0$. For any ${2\leq p < \infty}$ it holds that

$\displaystyle \boxed{\|f\|_{L^p([0,1])} \lesssim p^{1/2}\, \|S_{\mathcal{D}}f\|_{L^p([0,1])}.} \ \ \ \ \ \ (\text{CWW}_1)$

An important point about the above inequality is the behaviour of the constant in the Lebesgue exponent ${p}$, which is sharp. This can be seen by taking a “lacunary” function ${f}$ (essentially one where $\mathbf{D}_jf = a_j \in \mathbb{C}$, a constant) and randomising the signs using Khintchine’s inequality (indeed, ${p^{1/2}}$ is precisely the asymptotic behaviour of the constant in Khintchine’s inequality; see Exercise 5 in the 2nd post on Littlewood-Paley theory).
It should be remarked that the inequality extends very naturally and with no additional effort to higher dimensions, in which $[0,1]$ is replaced by the unit cube $[0,1]^d$ and the dyadic intervals are replaced by the dyadic cubes. We will only be interested in the one-dimensional case here though.