# 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.

# Marcinkiewicz-type multiplier theorem for q-variation (q > 1)

Not long ago we discussed one of the main direct applications of the Littlewood-Paley theory, namely the Marcinkiewicz multiplier theorem. Recall that the single-variable version of this theorem can be formulated as follows:

Theorem 1 [Marcinkiewicz multiplier theorem]: Let ${m}$ be a function on $\mathbb{R}$ such that

1. $m \in L^\infty$
2. for every Littlewood-Paley dyadic interval $L := [2^k, 2^{k+1}] \cup [-2^{k+1},-2^k]$ with $k \in \mathbb{Z}$

$\displaystyle \|m\|_{V(L)} \leq C,$

where $\|m\|_{V(L)}$ denotes the total variation of ${m}$ over the interval $L$.

Then for any ${1 < p < \infty}$ the multiplier ${T_m}$ defined by $\widehat{T_m f} = m \widehat{f}$ for functions $f \in L^2(\mathbb{R})$ extends to an $L^p \to L^p$ bounded operator,

$\displaystyle \|Tf\|_{L^p} \lesssim_p (\|m\|_{L^\infty} + C) \|f\|_{L^p}.$

You should also recall that the total variation $V(I)$ above is defined as

$\displaystyle \sup_{N}\sup_{\substack{t_0, \ldots, t_N \in I : \\ t_0 < \ldots < t_N}} \sum_{j=1}^{N} |m(t_j) - m(t_{j-1})|,$

and if ${m}$ is absolutely continuous then ${m'}$ exists as a measurable function and the total variation over interval $I$ is given equivalently by $\int_{I} |m'(\xi)|d\xi$. We have seen that the “dyadic total variation condition” 2.) above is to be seen as a generalisation of the pointwise condition $|m'(\xi)|\lesssim |\xi|^{-1}$, which in dimension 1 happens to coincide with the classical differential Hörmander condition (in higher dimensions the pointwise Marcinkiewicz conditions are of product type, while the pointwise Hörmander(-Mihklin) conditions are of radial type; see the relevant post). Thus the Marcinkiewicz multiplier theorem in dimension 1 can deal with multipliers whose symbol is somewhat rougher than being differentiable. It is an interesting question to wonder how much rougher the symbols can get while still preserving their $L^p$ mapping properties (or maybe giving up some range – recall though that the range of boundedness for multipliers must be symmetric around 2 because multipliers are self-adjoint).

Coifman, Rubio de Francia and Semmes came up with an answer to this question that is very interesting. They generalise the Marcinkiewicz multiplier theorem (in dimension 1) to multipliers that have bounded ${q}$-variation with ${q}$ > 1. Let us define this quantity rigorously.

Definition: Let $q \geq 1$ and let $I$ be an interval. Given a function $f : \mathbb{R} \to \mathbb{R}$, its ${q}$-variation over the interval ${I}$ is

$\displaystyle \|f\|_{V_q(I)} := \sup_{N} \sup_{\substack{t_0, \ldots t_N \in I : \\ t_0 < \ldots < t_N}} \Big(\sum_{j=1}^{N} |f(t_j) - f(t_{j-1})|^q\Big)^{1/q}$

Notice that, with respect to the notation above, we have $\|m\|_{V(I)} = \|m\|_{V_1(I)}$. From the fact that $\|\cdot\|_{\ell^q} \leq \|\cdot \|_{\ell^p}$ when $p \leq q$ we see that we have always $\|f\|_{V_q (I)} \leq \|f\|_{V_p(I)}$, and therefore the higher the ${q}$ the less stringent the condition of having bounded ${q}$-variation becomes (this is linked to the Hölder regularity of the function getting worse). In particular, if we wanted to weaken hypothesis 2.) in the Marcinkiewicz multiplier theorem above, we could simply replace it with the condition that for any Littlewood-Paley dyadic interval $L$ we have instead $\|m\|_{V_q(L)} \leq C$. This is indeed what Coifman, Rubio de Francia and Semmes do, and they were able to show the following:

Theorem 2 [Coifman-Rubio de Francia-Semmes, ’88]: Let $q\geq 1$ and let ${m}$ be a function on $\mathbb{R}$ such that

1. $m \in L^\infty$
2. for every Littlewood-Paley dyadic interval $L := [2^k, 2^{k+1}] \cup [-2^{k+1},-2^k]$ with $k \in \mathbb{Z}$

$\displaystyle \|m\|_{V_q(L)} \leq C.$

Then for any ${1 < p < \infty}$ such that ${\Big|\frac{1}{2} - \frac{1}{p}\Big| < \frac{1}{q} }$ the multiplier ${T_m}$ defined by $\widehat{T_m f} = m \widehat{f}$ extends to an $L^p \to L^p$ bounded operator,

$\displaystyle \|Tf\|_{L^p} \lesssim_p (\|m\|_{L^\infty} + C) \|f\|_{L^p}.$

The statement is essentially the same as before, except that now we are imposing control of the ${q}$-variation instead and as a consequence we have the restriction that our Lebesgue exponent ${p}$ satisfy ${\Big|\frac{1}{2} - \frac{1}{p}\Big| < \frac{1}{q} }$. Taking a closer look at this condition, we see that when the variation parameter is $1 \leq q \leq 2$ the condition is empty, that is there is no restriction on the range of boundedness of $T_m$: it is still the full range ${1}$ < ${p}$ < $\infty$, and as ${q}$ grows larger and larger the range of boundedness restricts itself to be smaller and smaller around the exponent $p=2$ (for which the multiplier is always necessarily bounded, by Plancherel). This is a very interesting behaviour, which points to the fact that there is a certain dichotomy between variation in the range below 2 and the range above 2, with $2$-variation being the critical case. This is not an isolated case: for example, the Variation Norm Carleson theorem is false for ${q}$-variation with ${q \leq 2}$; similarly, the Lépingle inequality is false for 2-variation and below (and this is related to the properties of Brownian motion).

# Representing points in a set in positional-notation fashion (a trick by Bourgain): part II

This is the second and final part of an entry dedicated to a very interesting and inventive trick due to Bourgain. In part I we saw a lemma on maximal Fourier projections due to Bourgain, together with the context it arises from (the study of pointwise ergodic theorems for polynomial sequences); we also saw a baby version of the idea to come, that we used to prove the Rademacher-Menshov theorem (recall that the idea was to represent the indices in the supremum in their binary positional notation form and to rearrange the supremum accordingly). Today we finally get to see Bourgain’s trick.

Before we start, recall the statement of Bourgain’s lemma:

Lemma 1 [Bourgain]: Let $K$ be an integer and let $\Lambda = \{\lambda_1, \ldots, \lambda_K \}$ a set of ${K}$ distinct frequencies. Define the maximal frequency projections

$\displaystyle \mathcal{B}_\Lambda f(x) := \sup_{j} \Big|\sum_{k=1}^{K} (\mathbf{1}_{[\lambda_k - 2^{-j}, \lambda_k + 2^{-j}]} \widehat{f})^{\vee}\Big|,$

where the supremum is restricted to those ${j \geq j_0}$ with $j_0 = j_0(\Lambda)$ being the smallest integer such that $2^{-j_0} \leq \frac{1}{2}\min \{ |\lambda_k - \lambda_{k'}| : 1\leq k\neq k'\leq K \}$.
Then

$\displaystyle \|\mathcal{B}_\Lambda f\|_{L^2} \lesssim (\log \#\Lambda)^2 \|f\|_{L^2}.$

Here we are using the notation $(\mathbf{1}_{[\lambda_k - 2^{-j}, \lambda_k + 2^{-j}]} \widehat{f})^{\vee}$ in the statement in place of the expanded formula $\int_{|\xi - \lambda_k| < 2^{-j}} \widehat{f}(\xi) e^{2\pi i \xi x} d\xi$. Observe that by the definition of $j_0$ we have that the intervals $[\lambda_k - 2^{-j_0}, \lambda_k + 2^{-j_0}]$ are disjoint (and $j_0$ is precisely maximal with respect to this condition).
We will need to do some reductions before we can get to the point where the trick makes its appearance. These reductions are the subject of the next section.

3. Initial reductions

A first important reduction is that we can safely replace the characteristic functions $\mathbf{1}_{[\lambda_k - 2^{-j}, \lambda_k + 2^{-j}]}$ by smooth bump functions with comparable support. Indeed, this is the result of a very standard square-function argument which was already essentially presented in Exercise 22 of the 3rd post on basic Littlewood-Paley theory. Briefly then, let $\varphi$ be a Schwartz function such that $\widehat{\varphi}$ is a smooth bump function compactly supported in the interval $[-1,1]$ and such that $\widehat{\varphi} \equiv 1$ on the interval $[-1/2, 1/2]$. Let $\varphi_j (x) := \frac{1}{2^j} \varphi \Big(\frac{x}{2^j}\Big)$ (so that $\widehat{\varphi_j}(\xi) = \widehat{\varphi}(2^j \xi)$) and let for convenience $\theta_j$ denote the difference $\theta_j := \mathbf{1}_{[-2^{-j}, 2^{-j}]} - \widehat{\varphi_j}$. We have that the difference

$\displaystyle \sup_{j\geq j_0(\Lambda)} \Big|\sum_{k=1}^{K} ((\mathbf{1}_{[\lambda_k - 2^{-j}, \lambda_k + 2^{-j}]} - \widehat{\varphi_j}(\cdot - \lambda_k)) \widehat{f})^{\vee}\Big|$

is an $L^2$ bounded operator with norm $O(1)$ (that is, independent of $K$). Indeed, observe that $\mathbf{1}_{[\lambda_k - 2^{-j}, \lambda_k + 2^{-j}]}(\xi) - \widehat{\varphi_j}(\xi - \lambda_k) = \theta_j (\xi - \lambda_k)$, and bounding the supremum by the $\ell^2$ sum we have that the $L^2$ norm (squared) of the operator above is bounded by

$\displaystyle \sum_{j \geq j_0(\Lambda)} \Big\|\sum_{k=1}^{K} (\theta_j(\cdot - \lambda_k)\widehat{f})^{\vee}\Big\|_{L^2}^2,$

where the summation in ${j}$ is restricted in the same way as the supremum is in the lemma (that is, the intervals $[\lambda_k - 2^{-j}, \lambda_k + 2^{-j}]$ must be pairwise disjoint). By an application of Plancherel we see that the above is equal to

$\displaystyle \sum_{k=1}^{K} \Big\| \widehat{f}(\xi) \Big[\sum_{j \geq j_0} \theta_j(\xi - \lambda_k) \Big]\Big\|_{L^2}^2;$

but notice that the functions $\theta_j$ have supports disjoint in ${j}$, and therefore the multiplier satisfies $\sum_{j\geq j_0} \theta_j(\xi - \lambda_k) \lesssim 1$ in a neighbourhood of $\lambda_k$, and vanishes outside such neighbourhood. A final application of Plancherel allows us to conclude that the above is bounded by $\lesssim \|f\|_{L^2}^2$ by orthogonality (these neighbourhoods being all disjoint as well).
By triangle inequality, we see therefore that in order to prove Lemma 1 it suffices to prove that the operator

$\displaystyle \sup_{j} \Big|\sum_{k=1}^{K} (\widehat{\varphi_j}(\cdot - \lambda_k) \widehat{f})^{\vee}\Big|$

is $L^2$ bounded with norm at most $O((\log \#\Lambda)^2)$.