fourier analysis - Why do Littlewood-Paley projections behave like iid random variables

There is a quantitative way to express the somewhat vague notion of "almost independence of the Littlewood-Paley projections".

Let $mathcal F_n$, $ninmathbb Z$, be the minimal $sigma$-algebra generated by the set $mathcal D_n$ of
dyadic cubes in $mathbb R^d$

$$mathcal D_n=left{prodlimits_{k=1}^{d}[m_k2^{-n},(m_k+1)2^{-n})|quad (m_1,dots,m_d)inmathbb Z^dright}.$$
Then for any locally integrable function $f$ on $mathbb R^d$, one may define the conditional
expectation $E_n(f)$ with respect to the filtration of $sigma$-algebras ${mathcal F_k| kinmathbb Z }$:
$$E_n(f)=sumlimits_{Qin mathcal D_n}chi_Q frac{1}{|Q|}int_Q f(x)dx.$$
It is not hard to check that the differences $D_n(f)=E_n(f)-E_{n-1}(f)$, $ninmathbb Z$,
define a martingale. This means that the family of Haar functions has the martingale property (and they indeed can be viewed as iid random variables).

Now, the Littlewood-Paley projections $Delta_n$ (and partial sums of Fourier series, in general) cannot be interpreted directly as conditional expectations. However, they do behave almost like the family of Haar functions. Roughly speaking, the families of projections ${Delta_k}_{kinmathbb Z}$ and ${D_j}_{jinmathbb Z}$ are almost biorthogonal.

Theorem. There exists a constant $C$ such that for every $k$, $jinmathbb Z$ the following estimate on the operator norm of $D_kDelta_j: L^2(mathbb R^n)to L^2(mathbb R^n)$ is valid

$$|D_kDelta_j|=|Delta_jD_k|leq C2^{-|j-k|}.$$

This result is relatively recent and is due to Grafakos and Kalton (see Chapter 5 of the book by Grafakos).

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