why is paraproduct or paradifferential calculus important in PDE theory? In the article https://www.baidu.com/link?url=W1BjGmDoZM8QkrV_Qd_26vzNhCJGWyfH79q5cn7q0QQxomVLtH7Fw_mApElkfCZUWiDcYjNhoLhMrGFEXtf4O_&wd=&eqid=a93906890002f93700000003577cbb98, it says that "...paraproducts have played a central role in analysis and PDEs..." In what sense and how does the development of the paradifferential calculus influence the modern analysis of PDEs? And how does the development of analysis of PDEs influence the paradifferential calculus theory?
I know one example, Danchin's work on the existence of smooth solutions for Navier-Stokes equations in critical spaces (Danchin R. Global existence in critical spaces for compressible Navier-Stokes equations[J]. Inventiones Mathematicae, 2000, 141(3): 579-614.). But I am not familiar with the other examples.
 A: 
How does the development of paradifferential calculus influence the modern analysis of PDEs?

You can get a quick glance by looking at Bony's 1981 paper with its 476 citations on MathSciNet. (Also, the few sentences after those which you quoted in the article you linked to gave already several broad applications that are extremely useful in PDEs.)
One item not on that list, however, is Hormander's 1990 paper on Nash-Moser Implicit Function Theorem, from which we can quote:

The Nash-Moser method contains constructions which very much resemble the dyadic decompositions which are central in the paradifferential calculus of Bony. In fact, we show that the Nash-Moser technique can often be replaced by elementary nonlinear functional analysis combined with the paradifferential calculus of Bony.

In terms of paraproducts, I think Tao's lecture notes give a fairly accurate account of how many PDE people actually think about paraproducts: as a particular way of exposing structures arising from interactions of different frequencies in a product; these structures can then be coupled with knowledge of the linearized evolution to get sharper estimates. 
