Paradifferential Calculus

Abstract

This third chapter constructs a version of paradifferential calculus that plays an essential role in the proof of the main result of the book. The symbol a(U;t, x, ξ) of a paradifferential operator on the circle is a function of the variables (x, ξ) belonging to ${{\mathbb {T}}^1}\times {\mathbb {R}}$, which depends on the space variable x (and on time t) only through the solution U itself of the water waves equations. For such a reason the symbol has a finite regularity with respect to the space variable x, and we define its Bony-regularization. The dependence of each symbol U → a(U;⋅) with respect to U—belonging to some functional space—admits an expansion in homogeneous components up to degree N. The goal of this chapter is to define such classes of symbols, study their Bony-Weyl quantization, and their symbolic calculus. We study also the paracomposition operator associated to a diffeomorphism on the circle.