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Paracomposition et opérateurs paradifferentiels. (Paracomposition and paradifferential operators). (French) Zbl 0596.47023
In a paper from 1981 [Ann. Sci. Éc. Norm. Super., IV Sér. 14, 209-246 (1981; Zbl 0495.35024)], J.-M. Bony introduces the notion of the paradifferential operator and the paramultiplication, used in the study of the nonlinear partial differential equations. In a similar context the author introduces and examines the properties of the ”paracomposition” \(X^*:{\mathcal D}'(\Omega_ 2)\to {\mathcal D}'(\Omega_ 1)\), where \(X:\Omega_ 1\to \Omega_ 2\) is a diffeomorphism of class \({\mathbb{C}}^{\rho +1}\) \((\rho >0)\), \(\Omega_ 1\) and \(\Omega_ 2\) are two open sets in \(R^ n\). Among other results, the existence of a linear operator \(X^*\) with the following properties is shown: \(X^*:H^ s_{loc}(\Omega_ 2)\to H^ s_{loc}(\Omega_ 1)\); \(X^*:C^{\sigma}_{loc}(\Omega_ 2)\to C^{\sigma}_{loc}(\Omega_ 1)\); \(u\circ X=X^*u+T_{u'\circ X}X+R\), where \(H^ s\) and \(C^{\rho}\) are the Sobolev and the Hölder function spaces respectively, \(R\in C_{loc}^{\rho +1+\epsilon}\) \((\epsilon =\inf (\sigma -1,\rho +1))\) if \(u\in C^{\sigma}_{loc}\) \((\sigma >1,\rho >0)\), and \(R\in H_{loc}^{r+1+\epsilon}\) \((\epsilon =\inf (s- (n/2)-1,r-(n/2)+1))\) if \(u\in H^ s_{loc}\) \((s>(n/2)+1,r>n/2)\), \(T_{u'\circ X}\) being the Bony’s paramultiplication and \(X^*u\) the new paracomposition.
Reviewer: I.Gottlieb

MSC:
47B38 Linear operators on function spaces (general)
35L75 Higher-order nonlinear hyperbolic equations
46F10 Operations with distributions and generalized functions
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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[1] Beals M, Spreading of singularities for a semi-linear wave equation 49 pp 275– (1982) · Zbl 0496.35059
[2] Beals M, Self-spreading and strength of singularities for solutions to semi-linear wave equations 118 pp 187– (1983) · Zbl 0522.35064
[3] Beals M, Propagation of singularities for hyperbolic pseudodifferential operators with non-smooth coefficients 35 pp 169– (1982) · Zbl 0482.35083
[4] Beals M, Microlocal regularity theorems for non-smooth pseudodifferential operators and applications to non linear problems 285 (1984) · Zbl 0562.35093
[5] Bony J-M, Calcul symbolique et propagation des singularités pour les eacute;quations aux dérivees partielles non-linéaires 14 pp 209– (1981)
[6] Bony J-M, Sem. Goulaouic-Meyer-Schwartz 14 (1979)
[7] Bony J-M, Sem. Goulaouic-Meyer-Schwartz 14 (1981)
[8] Bony J-M, Séminaire Goulaouic-Meyer–Schwartz 14 (1983)
[9] Boulkhemair A., Thèse de 3ème cycle (1984)
[10] Coifman R, Astèrisque 57 (1978)
[11] Lascar B, Singularités des solutions déquations aux dérivees partielles non-lineaires 287 pp 527– (1978)
[12] Leichtnam E, Front d1 onde d’une sous-variété propagation des singularités pour des équations aux dérivées partielles non linéaires
[13] Melrose R, Interaction of non-linear progressing waves (1984) · Zbl 0554.35085
[14] Meyer Y, Remarque sur un theoreme de J-M. Bony pp 1– (1981)
[15] Rauch J, Propagation of singularities for semi-linear hyperbolic equations in one space variable 111 pp 531– (1980)
[16] Rauch J, Jump discontinuities of semilinear strictly hyperbolic systems in two variables : creation and propagation 81 pp 203– (1981) · Zbl 0468.35064
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