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On paracomposition and change of variables in paradifferential operators

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Abstract

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 Alinhac (Commun Part Differ Equ 11(1):87–121, 1986). 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 that is a change of variable. We give a proof of the change of variables in paradifferential operators. Finally we study the cutoff defining paradifferential operators and it’s stability by successive composition. It is known that the cutoff becomes worse after each composition, we give a slightly refined version of the cutoffs proposed by Hörmander (Lectures on nonlinear hyperbolic differential equations, Springer, Berlin, 1997) for which give an optimal estimate on the support of the cutoff after composition.

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Notes

  1. The so called good unknown of Alinhac.

  2. This is not immediate from the definition but is a consequence of the fact that \(H^s(\Omega )\) can be seen as a quotient of \(H^s({\mathbb {R}}^d)\) by a closed subset, for a full presentation see [11].

  3. \(R_\omega \) is clearly open.

  4. In fact it can be treated in the more general frame of operators with singular symbols but this goes beyond the scope of this work.

  5. Part 3.3 point h, which can be found in pages 114–115.

  6. Clearly when there is no diffeomorphism hypothesis on \(\chi \) we can choose \(\chi :{\mathbb {R}}^d \rightarrow {\mathbb {R}}^{d'}\) with \(d\ne d'\) and have the same results but for clarity we chose to present the same dimensions.

  7. Note that this extra hypothesis is needed for the methods used to work and is not intrinsic to the problem. Also this hypothesis is immediately verified in the diffeomorphism case treated by Alinhac.

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Acknowledgements

I would like to express my sincere gratitude to my thesis advisor Thomas Alazard.

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  1. Duke University, Durham, USA

    Ayman Rimah Said

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A. R. Said wrote the full manuscript text.

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Said, A.R. On paracomposition and change of variables in paradifferential operators. J. Pseudo-Differ. Oper. Appl. 14, 25 (2023). https://doi.org/10.1007/s11868-023-00510-0

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