I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for.

For any $f \in L^2(\mathbb{R})$, let $\mathcal{Cf}:=\sup_{N \in \mathbb{Z}} \left\vert P_{-}(e^{iN\cdot}f)\right\vert$ be the maximal Carleson operator; where $P_{-}$ is the projection on negative Fourier spectrum $\{\xi <0\}$. The Carleson theorem essentially states that:

Theorem (Carleson).$|\{\mathcal{C}f > \lambda\}|_{L^2} \lesssim \lambda^{-2}\|f\|^2_{L^2}$.

In the modern proof of this theorem [I'm reading this], one instead studies the operators defined by $Q_{\xi}f:= \sum_{s \in T} \mathbf{1}_{\omega_s^+}(\xi) \langle f, \varphi _s\rangle \varphi _s$, where the notation is as follows:

- $T$ denotes the set of all tiles $I_s \times \omega_s$ such that $I_s, \omega_s$ are dyadic intervals, such that the area of the tile $I_s \times \omega_s$ is one.
- $\omega_s^+$ stands for the upper half of the interval, and the $\varphi_s$ are functions such that $\hat \phi_s$ have Fourier support inside $\omega_s^{-}$ (the lower half of the interval)

It is not difficult to pass from $Q_{\xi}$ to the Carleson operator, one can take averages and get that the operator: $$Q:=\lim_{Y \to \infty} \frac{1}{Y^2}\int_{[1,2] \times [0,Y]^2} Dil^2_{2^{-\lambda}}Tr_{-y}Mod_{-\xi}Q_{\xi} Mod_{\xi}Tr_yDil_{2^{\lambda}}^2d\lambda dy d\xi\,,$$ commutes with translations and dilations and its Kernel is made of functions with Fourier support lying on $\{\xi >0\}$, thus this operator is $P_{-}$.

My question is then:

How is it somewhat 'natural' to come up with the operators $Q_{\xi}$? How one can guess that such an operator has a similar behavior to that of $\mathcal C$?

I think there is some 'discretization' idea behind but I do not see how is it natural in any sense. Put in another way my question is: starting from $\mathcal C$ and $P_{-}$ how does one introduces the operators $Q_{\xi}$?

Does anybody has some good insights?