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.

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Weights theory basics, pt. II

Pdf version here: link.

This is the 2nd part of a post on basic weights theory. In the previous part I included definitions and fundamental facts, namely the nestedness of the {A_p} classes, the weighted inequalities for {M} and an extrapolation result for general operators. Refer to part I for notation and those results. This part includes instead a fundamental result on Calderón-Zygmund operators, together with inequalities relevant to the proof, and an elegant proof of Marcinkiewicz’s multiplier theorem as an application.

All of the following is taken from the excellent [Duo].

4. Calderón-Zygmund operators

This section is about the following result: if {T} is now a Calderón-Zygmund operator, then {T} is bounded on any {L^p (A_p)} for all {1<p<\infty}; moreover, if {p=1} then {T} is weakly bounded on {L^1 (A_1)}. This is exactly what one would’ve hoped for, and suggests that the weights theory is indeed one worth pursuing.

Recall a CZ-operator is an {L^2} bounded operator given by integration against a CZ-kernel {K}, defined on {\mathcal{S}(\mathbb{R}^d)} at least, i.e.

\displaystyle Tf(x) = \int{K(x,y)f(y)}\,dy,

with the properties (for some {\delta > 0})

  1. {|K(x,y)|\lesssim \frac{1}{|x-y|^n}};
  2. {|K(x,y)- K(x,y')|\lesssim \frac{|y-y'|^\delta}{(|x-y|+|x-y'|)^{d+\delta}}\qquad} for {|y-y'| \leq \frac{1}{2}\max\{|x-y|,|x-y'|\}};
  3. {|K(x,y)- K(x',y)|\lesssim \frac{|x-x'|^\delta}{(|x-y|+|x'-y|)^{d+\delta}}\qquad} for {|x-x'| \leq \frac{1}{2}\max\{|x-y|,|x'-y|\}}.

Then, let’s state formally

Theorem 6 Let {T} be a Calderón-Zygmund operator. Then {T} is bounded on all {L^p (A_p)} for {1<p<\infty}:

\displaystyle \int{|Tf|^p w} \lesssim \int{|f|^p w}\qquad \forall w \in A_p.

Moreover,

\displaystyle w\left(\{|Tf|>\lambda\}\right) \lesssim \frac{1}{\lambda}\int{|f|w}\qquad \forall w \in A_1.

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Weights theory basics, pt. I

Pdf version here: link.

Let’s work on {\mathbb{R}^d} for some fixed dimension {d}. In the following I’ll write {M} for the maximal function on uncentered cubes. In the following setting it won’t make a difference whether to work with this or with the usual one (defined on the centered balls), except in the constants; which I’ll throughout ignore thanks to my friend the wiggle notation and the fact that the dimension is fixed. In the proofs one has to resort alternatively to the dyadic, Hardy-Littlewood, centered versions of {M} but it really just amounts to technicalities. I’ll ignore these minor issues as the reader can easily see how they are overcome. When in doubt, refer to the excellent [Duo], which is where all of the following is taken from.

I’ve split the post in two for convenience, you can find part II here.

1. Definition of {A_p} and weighted inequalities for the maximal function {M}

Definition 1 ({A_p} weights) A locally integrable non negative function {w} is said to be in the {A_p} weight class if:

  • in case {p=1}, it holds that for any {x} and any cube {Q \ni x}

    \displaystyle \frac{w(Q)}{|Q|} \lesssim_w w(x) \ \ \ \ \ (1)

    (here {w(Q)} is the measure of {Q} w.r.t. measure {w(y)\,dy}, so {w(Q)=\int_{Q}{w}\,dy});

  • in case {p>1}, for any cube {Q} it must be

    \displaystyle \left(\frac{1}{|Q|}\int_{Q}{w}\right)\left(\frac{1}{|Q|}\int_{Q}{w^{1-p'}}\right)^{p-1}\lesssim_w 1. \ \ \ \ \ (2)

 

It is useful to refer to the quantity

\displaystyle [w]_{A_p} :=\sup_{Q}{\left(\frac{1}{|Q|}\int_{Q}{w}\right)\left(\frac{1}{|Q|}\int_{Q}{w^{1-p'}}\right)^{p-1}}

as the {A_p} constant of {w}.
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