Published in arXiv preprint, 2020

In this paper we revisit the hypothesis needed to define the “paracomposition” operator, an analogue to the classic pull-back operation in the low regularity setting, first introduced by S. Alinhac. More precisely we do so in two directions. First we drop the diffeomorphism hypothesis. Secondly we give estimates in global Sobolev and Zygmund spaces. Thus we fully generalize Bony’s classic paralinearasition Theorem giving sharp estimates for composition in Sobolev and Zygmund spaces. In order to prove that the new class of operations benefits of symbolic calculus properties when composed by a paradifferential operator, we discuss the pull-back of pseudodifferential and paradifferential operators which then become Fourier Integral Operators. In this discussion we show that those Fourier Integral Operators obtained by pull-back are pseudodifferential or paradifferential operators if and only if they are pulled-back by a diffeomorphism i.e a change of variable. We give a proof of the change of variables in paradifferential operators.

Finally we give a cutoff defining paradifferential operators that is stable by successive composition. This permits the definition and convergence in the class of paradifferential operators of series, for example the exponential of a paradifferential operator of negative order. This is the main new technical result of this paper.

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