On paracomposition and change of variables in Paradifferential operators

TL;DR

The paper generalizes Alinhac's paracomposition by dropping the diffeomorphism requirement and establishing global estimates in Sobolev spaces $H^s$ and Zygmund spaces $C^r_*$, showing $u \circ \chi = \chi^\star u + T_{u'(\chi)}\chi + \text{remainder}$ with controlled bounds. It develops a comprehensive pull-back theory for pseudodifferential and paradifferential operators, linking pull-backs to Fourier Integral Operators and proving that such pull-backs remain pseudodifferential/paradifferential precisely when the map is a change of variables; a formal change-of-variables formula for paradifferential operators is provided. A novel asymmetrical cutoff framework (Hörmander–Métivier cutoffs) is introduced to obtain sharp spectral localization under operator composition, including optimal support results. The results extend paracomposition to open subsets and establish robust estimates and functorial properties, with applications to low-regularity analyses such as water-wave problems. Collectively, the work unifies paracomposition, symbolic calculus, and variable-coefficient changes in a global, low-regularity setting, enabling sharper composition estimates in global function spaces.

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 S. Alinhac in [3]. 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 in [14] for which give an optimal estimate on the support of the cutoff after composition.

Figures (4)

• Figure 1: Hörmander's choice of cut-off function $(\psi_H^B)_{B>2}$.
• Figure 2: Métivier choice of cut-off function $(\psi^\epsilon_M)_{\epsilon<1}$.
• Figure 3: The choice of cut-off function $(\psi^{B_1,B_2,b})_{B_1>0,B_2>1,b>0}$, $d=1$.
• Figure 4: Admissible cut-off functions.

Theorems & Definitions (74)

• Definition 1.1
• Theorem 1.1
• Corollary 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 2.1: Littlewood-Paley decomposition
• Proposition 2.1: Bernstein's inequalities
• Proposition 2.2
• Definition 2.2: Singular support
• Definition 2.3: Zygmund spaces on $\mathbb R^d$