Question on Littlewood-Paley trichotomy

Question

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 $2^{k-1} \le |\xi| \le 2^{k+1}$?

Answer 1

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.