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