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abstract: 'We consider here pseudo-differential operators whose symbol $\sigma(x,\xi)$ is not infinitely smooth with respect to $x$. Decomposing such symbols into four -sometimes five- components and using tools of paradifferential calculus, we derive sharp estimates on the action of such pseudo-differential operators on Sobolev spaces and give explicit expressions for their operator norm in terms of the symbol $\sigma(x,\xi)$. We also study commutator estimates involving such operators, and generalize or improve the so-called Kato-Ponce and Calderon-Coifman-Meyer estimates in various ways.'
address: |
    MAB, Université Bordeaux I et CNRS, UMR 5466\
    351 Cours de la Libération,\
    33405 Talence Cedex, France.
author:
- David Lannes
title: Sharp estimates for pseudodifferential operators with symbols of limited smoothness and commutators
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pseudodifferential operators ,paradifferential calculus , commutator estimates

Introduction
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General setting and description of the results
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Among the widely known properties of pseudo-differential operators with symbol in Hormander’s class $S^m_{1,0}$, two are discussed in this paper. The first one concerns their action on Sobolev spaces, and the second one deals with the properties of commutators.

It is a classical result that for all $\sigma^1\in S^{m_1}_{1,0}$, the operator ${\mbox{\textnormal{Op}}}(\sigma^1)$ maps $H^{s+m_1}({{\mathbb R}}^d)$ into $H^s({{\mathbb R}}^d)$ for all $s\in{{\mathbb R}}$. Moreover, the proof 