# Question on Littlewood-Paley trichotomy

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

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.

• Thanks. I used to think $|\xi + \zeta| \le 2^{k+1} + 2^{l+1}$ and get stuck. I am now studying on the paradifferential calculus with Tao's notes. Could you provide some references to study this field? I work in the field of hyperbolic PDEs. Thank you!
– sam
May 7, 2016 at 5:06
• Tao's notes are quite good for an introduction. You can also look around on the web for Klainerman's Lecture Notes on Analysis for the intro class in Princeton (look for the 2008 or later versions); his website was unfortunately somewhat broken the last time I checked. If you can read French, Bony's original papers are rather approachable. Additionally, while I have not read it, Metivier's notes look promising. May 8, 2016 at 3:32
• Thank you, Wong. I have Klainerman's 2011 course note. I will study on Metivier's notes. Your kindly reply helps a lot!
– sam
May 8, 2016 at 8:32