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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