Paradifferential Calculus

  • Massimiliano Berti
  • Jean-Marc Delort
Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 24)


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.


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    Alinhac, S.: Paracomposition et opérateurs paradifférentiels. Commun. Partial Differ. Equ. 11(1), 87–121 (1986). MathSciNetCrossRefGoogle Scholar
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    Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. London Mathematical Society Lecture Note Series, vol. 268. Cambridge University Press, Cambridge (1999).

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Authors and Affiliations

  • Massimiliano Berti
    • 1
  • Jean-Marc Delort
    • 2
  1. 1.Department of MathematicsInternational School for Advanced Studies SISSATriesteItaly
  2. 2.LAGASorbonne Paris-Cité/University Paris 13VilletaneuseFrance

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