# 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.

# 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).

# Basic Littlewood-Paley theory III: applications

This is the last part of a 3 part series on the basics of Littlewood-Paley theory. Today we discuss a couple of applications, that is Marcinkiewicz multiplier theorem and the boundedness of the spherical maximal function (the latter being an application of frequency decompositions in general, and not so much of square functions – though one appears, but only for $L^2$ estimates where one does not need the sophistication of Littlewood-Paley theory).
Part I: frequency projections
Part II: square functions

7. Applications of Littlewood-Paley theory

In this section we will present two applications of the Littlewood-Paley theory developed so far. You can find further applications in the exercises (see particularly Exercise 22 and Exercise 23).

7.1. Marcinkiewicz multipliers

Given an ${L^\infty (\mathbb{R}^d)}$ function ${m}$, one can define the operator ${T_m}$ given by

$\displaystyle \widehat{T_m f}(\xi) := m(\xi) \widehat{f}(\xi)$

for all ${f \in L^2(\mathbb{R}^d)}$. The operator ${T_m}$ is called a multiplier and the function ${m}$ is called the symbol of the multiplier1. Since ${m \in L^\infty}$, Plancherel’s theorem shows that ${T_m}$ is a linear operator bounded in ${L^2}$; its definition can then be extended to ${L^2 \cap L^p}$ functions (which are dense in ${L^p}$). A natural question to ask is: for which values of ${p}$ in ${1 \leq p \leq \infty}$ is the operator ${T_m}$ an ${L^p \rightarrow L^p}$ bounded operator? When ${T_m}$ is bounded in a certain ${L^p}$ space, we say that it is an ${L^p}$multiplier.

The operator ${T_m}$ introduced in Section 1 of the first post in this series is an example of a multiplier, with symbol ${m(\xi,\tau) = \tau / (\tau - 2\pi i |\xi|^2)}$. It is the linear operator that satisfies the formal identity $T \circ (\partial_t - \Delta) = \partial_t$. We have seen that it cannot be a (euclidean) Calderón-Zygmund operator, and thus in particular it cannot be a Hörmander-Mikhlin multiplier. This can be seen more directly by the fact that any Hörmander-Mikhlin condition of the form ${|\partial^{\alpha}m(\xi,\tau)| \lesssim_\alpha |(\xi,\tau)|^{-|\alpha|} = (|\xi|^2 + \tau^2)^{-|\alpha|/2}}$ is clearly incompatible with the rescaling invariance of the symbol ${m}$, which satisfies ${m(\lambda \xi, \lambda^2 \tau) = m(\xi,\tau)}$ for any ${\lambda \neq 0}$. However, the derivatives of ${m}$ actually satisfy some other superficially similar conditions that are of interest to us. Indeed, letting ${(\xi,\tau) \in \mathbb{R}^2}$ for simplicity, we can see for example that ${\partial_\xi \partial_\tau m(\xi, \tau) = \lambda^3 \partial_\xi \partial_\tau m(\lambda\xi, \lambda^2\tau)}$. When ${|\tau|\lesssim |\xi|^2}$ we can therefore argue that ${|\partial_\xi \partial_\tau m(\xi, \tau)| = |\xi|^{-3} |\partial_\xi \partial_\tau m(1, \tau |\xi|^{-2})| \lesssim |\xi|^{-1} |\tau|^{-1} \sup_{|\eta|\lesssim 1} |\partial_\xi \partial_\tau m(1, \eta)|}$, and similarly when ${|\tau|\gtrsim |\xi|^2}$; this shows that for any ${(\xi, \tau)}$ with ${\xi,\tau \neq 0}$ one has

$\displaystyle |\partial_\xi \partial_\tau m(\xi, \tau)| \lesssim |\xi|^{-1} |\tau|^{-1}.$

This condition is comparable with the corresponding Hörmander-Mikhlin condition only when ${|\xi| \sim |\tau|}$, and is vastly different otherwise, being of product type (also notice that the inequality above is compatible with the rescaling invariance of ${m}$, as it should be).