In proving the product estimate, we need the Littlewood-Paley trichotomy. See <http://www.math.ucla.edu/~tao/254a.1.01w/notes3.ps>.


In the decomposition
$$P\_k (fg)=\sum\_{k',k''\in Z} P\_k (P\_{k'} f P\_{k''} g)$$


The relation is obtained in analysis of the support of $D'=2^{k'-1} \le |\xi| \le 2^{k'+1}$,$D''=2^{k''-1} \le |\xi| \le 2^{k''+1}$. The note says $P\_{k'} f P\_{k''}g$ has Fourier support in $D'+D''$, what does it mean? I think it seems one need convolution to get the support for $|\xi|$. But I am unable to carry it out and derive the decomposition. How could we choose $k' \le k-5$ and find $k'' \in [k-3,k+3]$ to be the intersection region with $2^{k-1} \le |\xi| \le 2^{k+1}$?



Question on Littlewood-Paley trichotomy
1. The set $D' + D''$ is the set $\{ \xi \in \mathbb{R}^n: \xi = \xi' + \xi'', \xi'\in D', \xi''\in D''\}$.
2. It is an elementary lemma to prove that if 
$$ \mathrm{supp}(f) \subset D', \mathrm{supp}(g)\subset D'' $$
then 
$$ \mathrm{supp}(f \* g) \subset D' + D'' $$
(I leave this as an exercise, but I note that you will use the statement "if $\xi \not\in D' + D''$, then whenever $\xi = \xi\_1 + \xi\_2$ we have $f(\xi\_1) g(\xi\_2) = 0$.")


It may help to visualize this by setting $f$ and $g$ to be the characteristic functions of some subsets of the plane and drawing out on a piece of paper explicitly the supports of $f$, $g$, and $f\*g$.
3. By various applications of the triangle inequality you have, for 
$$ 2^{k-1} \leq |\xi| \leq 2^{k+1}, 2^{\ell - 1} \leq |\zeta| \leq 2^{\ell+1} $$
that
$$ |\xi + \zeta| \leq 2^{\max(k,\ell) + 2} $$
$$ |\zeta + \xi| \geq 2^{\ell - 1} - 2^{k + 1} $$
(Notice that the second inequality is only useful if $\ell - 1 > k+1$, in which case we can get $\ldots \geq 2^{\ell - 2}$.) The trichotomy principle arises by finding combinations of $k',k''$ such that, for example, 
$$ 2^{\max(k',k'') + 2} < 2^{k - 1} $$
to say that $P\_k (P\_{k'}f P\_{k''}g) = 0$. What's left are the terms in the trichotomy.


