where to find a proof of Littlewood-Paley theorem by Khintchine's inequality I heard that one application of Khintchine's inequality is the proof of Littlewood-Paley theorem. I only know a proof of Littlewood-Paley theorem using vector version of Calderon-Zygmund theorem. Can anyone provide me a reference (e.g. a textbook) containing a proof of Littlewood-Paley theorem by using Khintchine's inequality?
Remark: I don't write down the statements of the theorems mentioned above since only experts know the reference. I don't need a proof written here since it can be very lengthy. I just need a reference book. 
 A: The proof, assuming you know the requisite harmonic analysis, is not that long, but I have not been successful in finding a proper reference. So I produce the proof here. I am guided by the comments of 
Schlag in his paper on the "distorted Fourier Transform" and the wikipedia page on Khintchine's inequality.
Reserve the notation $|\cdot|_2$ for the $\ell^2(\mathbb Z)$ norm. Setting $\epsilon_j, j\in\mathbb Z$ to be iid $\text{Unif}(\{-1,+1\})$ random variables, the version of Khintchine's inequality we need is (the left-hand side of)
$$ |(a_j)_{j\in\mathbb Z}|_2 \lesssim_p\left[ \mathbb  E\left| \sum_{j\in\mathbb Z}\epsilon_j a_j\right|^p\right]^{1/p} \lesssim_p  |(a_j)_{j\in\mathbb Z}|_2.$$
Setting $a_j = P_j f$ so that $(a_j)_{j\in\mathbb Z} = Sf$ is the Littlewood-Paley square function, ($P_j$ is a multiplier with symbol $\psi(2^{-j} \xi)$) we have for a.e. $x$
$$ |Sf|_2 \lesssim_p\left[ \mathbb  E\left| \sum_{j\in\mathbb Z}\epsilon_j P_j f\right|^p\right]^{1/p} .$$
Taking $L^p$ norms in $x$ and applying Fubini,
$$ \| |Sf|_2 \|_p \lesssim \left[\mathbb E \left\|\sum_{j\in\mathbb Z}\epsilon_j P_j f\right\|_p^p \right]^{1/p}.$$ 
Now one checks that the symbol $\mu$ of the Fourier multiplier $\sum_{j\in\mathbb Z}\epsilon_j {P_j}$,
$$ \mu  = \sum_{j\in\mathbb Z}\epsilon_j {\psi(2^{-j}\xi)}  $$
satisfies the conditions of Mikhlin's multiplier theorem, independently of the signs $\epsilon_j$, namely that
$$ \mu \in C^\infty(\mathbb R^n\setminus\{0\}),\\ |\nabla^k \mu(\xi)| \le \frac{C_k}{|\xi|^k} ,\quad \xi\neq 0, \quad 0\le k\le  n2+1. $$
This gives
$$ \left\|\sum_{j\in\mathbb Z}\epsilon_j P_j f\right\|_p \lesssim_p \|f\|_p$$
where the constant implied by the notation $\lesssim_p$ is independent of $\epsilon_j$. Plugging into the earlier line therefore makes the $\mathbb E[\cdot]$ vanish,
$$ \| |Sf|_2 \|_p \lesssim_p \|f\|_p,$$
and one gets the reverse inequality by the same duality argument you used in the vector Calderon-Zygmund argument (for example, see Tao's MA254A Winter 2001 notes here.)
