the action on Sobolev (and Zygmund) spaces of the four components into which each symbol is decomposed, and give precise estimate on the operator norm. These results generalize classical results of paraproduct theory and are in the spirit of [@Marschall] and especially [@Yamazaki1] (but the estimate we give here are different from the ones given in this latter reference). Gathering the estimates obtained on each component, we obtain a tame estimate on the action of the operator associated to the full symbol $\sigma(x,\xi)$.\
Section \[sectcomm\] is devoted to the study of commutator estimates. We first give in Prop. \[lemmIII1\] precise estimates for Meyer’s well-known result on the symbolic calculus for paradifferential operators. In Section \[sectfour\], we address the case of commutators between a Fourier multiplier $\sigma^1(D)$ and a pseudo-differential operator $\sigma^2(x,D)$; we study some particular cases, including the case when $\sigma^2(x,\xi)=\sigma^2(x)$ is a function. The case when $\sigma^1(x,D)$ is a pseudo-differential operator (and not only a Fourier multiplier) is then addressed in Section \[sectfin\].

Throughout this paper, we use the following notations.\
[**Notations.**]{} [**i.**]{} For all $a,b\in{{\mathbb R}}$, we write $a\wedge b:=\max\{a,b\}$;\
[**ii.**]{} For all $a\in {{\mathbb R}}$, we write $a_+:=\max\{a,0\}$, while $[a]$ denotes the biggest integer smaller than $a$;\
